Differential Equations Questions

We provide differential equations practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on differential equations skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

List of differential equations Questions

Question No Questions Class
1 ( left(x^{2}-y^{2}right) frac{d y}{d x}=x y . ) Solve the above
equation
A ( cdot e^{-x^{2} / 2 y^{2}}=k x )
B . ( e^{-x^{2} / 2 y^{2}}=k y )
C ( cdot e^{x^{2} / 2 y^{2}}=k x )
D cdot ( e^{x^{2} / 2 y^{2}}=k y )
12
2 Eliminate the arbitrary constants and obtain the differential equation satisfied by it: ( boldsymbol{y}=left(frac{boldsymbol{a}}{boldsymbol{x}^{2}}right)+boldsymbol{b} boldsymbol{x} )
A ( cdot x^{2} y^{prime prime}+2 x y^{prime}-2 y=0 )
B . ( x^{2} y^{prime prime}+2 x^{2} y^{prime}-2 y=0 )
C ( cdot x^{2} y^{prime prime}+2 x y^{prime}+2 x y=0 )
D. ( x^{2} y^{prime prime}-2 x y^{prime}+2 x y=0 )
12
3 A population grows at the rate of ( 5 % ) per year. Then the population will be doubled at
A. ( 10 log 2 ) years
B. 20 log 2 years
c. ( 30 log 2 ) years
D. ( 40 log 2 ) years
12
4 The order and degree of the differential
equation of all parabola whose axis is ( x ) axis
( A cdot 2, )
B. 2,
( c cdot 1,2 )
D. 1,1
12
5 Solution of the different equation,
( y d x-x d y+x y^{2} d x=0 ) can be
A ( cdot 2 x+x^{2} y=lambda y )
в. ( 2 y+y^{2} x=lambda y )
c. ( 2 y-y^{2} x=lambda y )
D. none of these
12
6 The solution of ( frac{d y}{d x}+frac{x^{2}}{y^{2}}=0 ) is:
A ( cdot x^{2}+y^{2}=c )
B . ( x^{2}-y^{2}=c )
c. ( x^{3}-y^{3}=c )
D. ( x^{3}+y^{3}=c )
12
7 The degree of the differential equation
( boldsymbol{x}=mathbf{1}+frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+frac{mathbf{1}}{mathbf{2 !}}left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}+ )
( frac{1}{3 !}left(frac{d y}{d x}right)^{3}+dots dots ) is?
( A cdot 3 )
B.
c. Not defined
D. None of these
12
8 The solution of the differential equation ( y frac{d y}{d x}=xleft[frac{y^{2}}{x^{2}}+frac{phileft(frac{y^{2}}{x^{2}}right)}{phi^{prime}left(frac{y^{2}}{x^{2}}right)}right] ) is (where, ( c )
is a constant)
( ^{mathrm{A}} cdot phileft(frac{y^{2}}{x^{2}}right)=c x )
в. ( _{x phi}left(frac{y^{2}}{x^{2}}right)=c )
( ^{mathrm{c}} cdot phileft(frac{y^{2}}{x^{2}}right)=c x^{2} )
D. ( x^{2} phileft(frac{y^{2}}{x^{2}}right)=c )
12
9 Find the order of the differential
equation:
( log frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}=left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{3}+boldsymbol{x} )
12
10 The solution of the differential equation
( boldsymbol{x} boldsymbol{y}^{2} boldsymbol{d} boldsymbol{y}-left(boldsymbol{x}^{3}+boldsymbol{y}^{3}right) boldsymbol{d} boldsymbol{x}=mathbf{0} ) is
A ( cdot y^{3}=3 x^{3}+c )
B . ( y^{3}=3 x^{3} log (c x) )
c. ( y^{3}=3 x^{3}+log (c x) )
D. ( y^{3}+3 x^{3}+log (c x) )
12
11 Find the differential equation of family
of curves ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}}(boldsymbol{A} cos boldsymbol{x}+boldsymbol{B} sin boldsymbol{x}) )
where ( A ) and ( B ) are arbitrary constants.
12
12 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) if ( boldsymbol{y}=boldsymbol{x}^{2}-boldsymbol{2}^{sin boldsymbol{x}} ) 12
13 The D.E whose solution is ( y=frac{c}{x} ) is:
A. ( y_{1}+x y=0 )
в. ( y_{1}=x y )
c. ( x cdot d y+y cdot d x=0 )
D. ( x . d y-y . d x=0 )
12
14 21. If(2+ sinx)
+ (y+1) cos x=0 and y(0) = 1, then
dx
is equal to :
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12
15 Solve:
¡) ( boldsymbol{y}=boldsymbol{e}^{2 x}(boldsymbol{a}+boldsymbol{b} boldsymbol{x}) quad boldsymbol{y}= )
( e^{x}(a cos x+b sin x) )
ii) Form the differential equation of the family of circle touching the y-axis at origin.
12
16 The differential equation for the family of curves ( mathbf{x}^{2}+mathbf{y}^{2}-mathbf{2} mathbf{a y}=mathbf{0}, ) where ( boldsymbol{a} ) is
an arbitrary constant is:
A ( cdot 2left(mathrm{x}^{2}-mathrm{y}^{2}right) mathrm{y}^{prime}=x y )
B ( cdot 2left(mathrm{x}^{2}+mathrm{y}^{2}right) mathrm{y}^{prime}=x y )
C ( cdotleft(x^{2}-y^{2}right) y^{prime}=2 x y )
D. ( left(x^{2}+y^{2}right) y^{prime}=2 x y )
12
17 Find the solution of the differential
equation ( (x log x) frac{d y}{d x}+y= )
( 2 x log x,(x geq 1) )
12
18 The solution of the differential equation ( sec ^{2} x cdot tan y d x+sec ^{2} y cdot tan x d y=0 )
is
A. ( tan x cdot cot y=C )
B. ( cot x cdot tan y=C )
( mathbf{c} cdot tan x cdot tan y=C )
( mathbf{D} cdot sin x cdot cos y=C )
12
19 Solve the differential equation ( left(x^{2}-right. ) ( left.boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{x}-boldsymbol{x} boldsymbol{y} boldsymbol{d} boldsymbol{y}=mathbf{0} )
A ( cdot y^{2}left(x^{2}-2 y^{2}right)=k )
B . ( x^{2}left(x^{2}-2 y^{2}right)=k )
C ( cdot y^{2}left(2 x^{2}-y^{2}right)=k )
D. ( x^{2}left(2 x^{2}-y^{2}right)=k )
12
20 The differential equation whose solution is Ax? + By = 1
where A and B are arbitrary constants is of [2006]
(a) second order and second degree
(b) first order and second degree
(c) first order and first degree
(d) second order and first degree
12
21 Find the differential equation of ( boldsymbol{y}= )
( boldsymbol{a} e^{3 x}+boldsymbol{b} e^{3 x} )
12
22 The degree of the differential equation ( boldsymbol{y}(boldsymbol{x})=mathbf{1}+frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+frac{mathbf{1}}{mathbf{1 . 2}}left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}+ )
( frac{1}{1.2 .3}left(frac{d y}{d x}right)^{3}+dots . . frac{1}{1.2 .3 dots dots . n}left(frac{d y}{d x}right)^{n} )
is.
( A cdot 2 )
B. 3
( c cdot 1 )
( D )
12
23 The degree and order of the differential equation ( left[1+left(frac{d y}{d x}right)^{3}right]^{7 / 3}=7left(frac{d^{2} y}{d x^{2}}right) )
respectively are:
A. 3 and 7
B. 3 and 2
( c .7 ) and 3
D. 2 and 3
12
24 Solve the following differential equation ( frac{d y}{d x}=1-cos x ) 12
25 Evaluate ( left(1-x^{2}right) frac{d y}{d x}-x y=1 ) 12
26 Find the differential equation of the family of curve represented by
( c(y+c)^{2}+x^{3}=0 )
12
27 The differential equation whose solution is ( y^{2}=3 a y-x^{3} ) is
A ( cdotleft(x^{3}-y^{2}right) frac{d y}{d x}=3 x^{2} y )
B. ( left(x^{3}-y^{2}right) frac{d y}{d x}=3 x y )
c. ( left(x^{3}-yright) frac{d y}{d x}=3 x y^{2} )
D. ( left(y^{2}-x^{3}right) frac{d y}{d x}=3 x y )
12
28 the general solution of differential
equation ( boldsymbol{x}^{4} frac{d y}{d x}+boldsymbol{x}^{3} boldsymbol{y}+operatorname{cosec} boldsymbol{x} boldsymbol{y}=mathbf{0} )
is
A ( cdot 2 cos (x y)+frac{1}{x^{2}}=0 )
B. ( 2 cos (x y)+frac{1}{y^{2}}=0 )
( c cdot 2 sin y+frac{1}{x^{2}}=c )
D. ( 2 sin (x y)+frac{1}{y^{2}}=c )
12
29 The D.E whose solution is ( y=A sin 2 x+B ) ( cos 2 x ) given as:
A ( cdot y_{2}=4 y )
B . ( y_{2}+3 y=0 )
( mathbf{c} cdot y_{2}+y=0 )
D . ( y_{2}+4 y=0 )
12
30 ( x=tan left(frac{1}{a} log yright) ) prove :
( left(1+x^{2}right) y_{2}+(2 x-a) y_{1}=0 )
12
31 The D.E whose solution ( y=A e^{5 x}+ )
( B e^{-2 x} ) is
A ( cdot y_{2}-3 y_{1}-10 y=0 )
в. ( y_{2}+3 y_{1}-10 y=0 )
c. ( y_{2}+3 y_{1}+10 y=0 )
D. ( y_{2}-3 y_{1}+10 y=0 )
12
32 The number of arbitrary constant in the particular solution of a differential equation is
( A cdot 3 )
B. 4
c. infinite
D. zero
12
33 The degree of tbe differential equation whose primitive is ( c^{2}+2 c y+a^{2}- )
( x^{2}=0, ) where ( c ) is an arbitrary and a is
definite constant is:
12
34 If ( m ) be the slope of a tangent to the
curve ( e^{y}=left(1+x^{2}right) ) then
A ( .|m|>1 )
B. ( m<1 )
c. ( |m|<1 )
D. ( |m| leq 1 )
12
35 The solution of ( left(x^{3}-2 y^{3}right) d x+ )
( 3 x y^{2} d y=0 ) is:
A ( cdot x^{3}-y^{3}=c x^{2} )
в. ( x^{3}=y^{3} )
c. ( x^{3}-y^{3}=c x )
D. ( x^{3}+y^{3}=c x^{2} )
12
36 The order of the differential equation ( left(frac{d y}{d x}right)^{2}+left(frac{d y}{d x}right)-sin ^{2} y=0 ) 12
37 Solve the differential equation: ( y d x- )
( boldsymbol{x} boldsymbol{d} boldsymbol{y}+boldsymbol{3} boldsymbol{x}^{2} boldsymbol{y}^{2} boldsymbol{e}^{boldsymbol{x}^{3}} boldsymbol{d} boldsymbol{x} )
A ( cdot frac{x}{y}=e^{x^{3}}+c )
В. ( -frac{x}{y}=e^{x^{-3}}+c )
c. ( -frac{x}{y}=e^{x^{3}}+c )
D. ( frac{x}{y}=e^{x^{-3}}+c )
12
38 Form the differential equation of the family of circles touching the X-axis at the origin. 12
39 Solve the equation:
( left(x^{2}+3 x y+y^{2}right) d x-x^{2} d y=0, ) given
that ( y=0 ) and ( x=1 )
12
40 The solution of ( x cos ^{2} y(d x)+ ) ( tan y(d y)=0 ) is:
( mathbf{A} cdot x^{2}+sec ^{2} y=c )
B . ( x^{2}+c o t^{2} y=c )
C ( cdot x^{2}+sin ^{2} y=c )
D. ( x^{2}+cos ^{2} y=c )
12
41 Solve the following systems of linear equations
( boldsymbol{x}+boldsymbol{y}=mathbf{5} )
( boldsymbol{y}+boldsymbol{z}=mathbf{3} )
( boldsymbol{x}+boldsymbol{z}=boldsymbol{4} )
12
42 Q Type your question.
1. ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}}+mathbf{1} quad: boldsymbol{y}^{prime prime}-boldsymbol{y}=mathbf{0} )
2. ( boldsymbol{y}=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{C} quad quad: boldsymbol{y}^{prime}-boldsymbol{2} boldsymbol{x}- )
( mathbf{2}=mathbf{0} )
3. ( boldsymbol{y}=cos boldsymbol{x}+boldsymbol{C} quad: quad boldsymbol{y}^{prime}+ )
( sin x=0 )
4. ( y=sqrt{1+x^{2}} )
( : boldsymbol{y}^{prime}= )
( frac{x y}{1+x^{2}} )
( mathbf{5} . boldsymbol{y}=boldsymbol{A} boldsymbol{x} quad quad: quad boldsymbol{x} boldsymbol{y}= )
( boldsymbol{y}(boldsymbol{x} neq mathbf{0}) )
6. ( boldsymbol{y}=boldsymbol{x} sin boldsymbol{x} )
( : boldsymbol{x} boldsymbol{y}=boldsymbol{y}+ )
( boldsymbol{x} sqrt{boldsymbol{x}^{2} boldsymbol{y}^{2}}(boldsymbol{x} neq boldsymbol{0} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{x}>boldsymbol{y} operatorname{or} boldsymbol{x}<boldsymbol{y}) )
7. ( boldsymbol{x} boldsymbol{y}=log boldsymbol{y}+boldsymbol{C} quad quad: boldsymbol{y}^{prime}= )
( frac{y^{2}}{1-x y}(x y neq 1) )
8. ( y-cos y=x )
( (y sin y+ )
( cos boldsymbol{y}+boldsymbol{x}) boldsymbol{y}^{prime}=boldsymbol{y} )
9. ( x+y=tan ^{-1} y )
( : quad y^{2} y^{prime}+ )
( boldsymbol{y}^{2}+mathbf{1}=mathbf{0} )
10. ( boldsymbol{y}=sqrt{boldsymbol{a}^{2}-boldsymbol{x}^{2}} boldsymbol{x} epsilon(-boldsymbol{a}, boldsymbol{a}): boldsymbol{x}+ )
( boldsymbol{y} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=mathbf{0}(boldsymbol{y} neq mathbf{0}) )
12
43 The order of the differential equation is
( A cdot 3 )
B. 2
( c cdot 1 )
D. None of these
12
44 Which of the following differential equation is linear?
A ( cdot frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}+2 y=0 )
B. ( frac{d^{2} y}{d x^{2}}+y frac{d y}{d x}+x=0 )
c. ( frac{d^{2} y}{d x^{2}}+frac{y}{x}+sin y=x^{2} )
D. ( (1+x) frac{d y}{d x}-x y=1 )
12
45 Solve the differential equation ( boldsymbol{x}+ )
( y frac{d y}{d x}=2 y )
12
46 solve
( x d y-y d x=sqrt{x^{2}+y^{2} d x} )
12
47 Solution of ( sqrt{1+x^{2}+y^{2}+x^{2} y^{2}}+ )
( x y frac{d y}{d x}=0, ) is:
A ( cdot log left(frac{x}{1+sqrt{1+x^{2}}}right)+sqrt{1+x^{2}}+sqrt{1+y^{2}}=c )
B ( cdot log left(frac{x}{sqrt{1+x^{2}}}right)+sqrt{1-x^{2}}+sqrt{1+y^{2}}=c )
( c cdot log left(frac{x}{sqrt{1+x^{2}}}right)=c )
D ( cdot log (sqrt{1+x^{2}}-sqrt{1+y^{2}})+log left(frac{x}{sqrt{1+x^{2}}}right)=c )
12
48 Solve: ( boldsymbol{y} boldsymbol{d} boldsymbol{x}+left(boldsymbol{x}-boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{y}=mathbf{0} ) 12
49 The general solution of differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}+boldsymbol{y}}{boldsymbol{x}-boldsymbol{y}} ) is 12
50 Determine the order and degree(if defined) of the following differential equation.
( boldsymbol{y}^{prime prime}+left(boldsymbol{y}^{prime}right)^{2}+mathbf{2} boldsymbol{y}=mathbf{0} )
12
51 The differential equation of the family of straight lines whose slope is equal to ( y ) intercept.
A ( cdot(x+1) frac{d y}{d x}-y=0 )
B. ( left(x+1 frac{d y}{d x}+y=0right. )
c. ( frac{d y}{d x}=frac{x-1}{y-1} )
D. ( frac{d y}{d x}=frac{x+1}{y+1} )
12
52 ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{y} cos boldsymbol{x}=boldsymbol{y}^{boldsymbol{n}} sin boldsymbol{2} boldsymbol{x} )
( mathbf{A} cdot frac{1}{y^{n+1}}=2 sin x-frac{2}{1-n}+c e^{(n-1) sin x} )
B. ( frac{1}{y^{n-1}}=2 sin x+frac{2}{1-n}+c e^{(n-1) sin x} x )
C ( frac{1}{y^{n-1}}=2 sin x-frac{2}{1-n}+c e^{(n-1) sin x} x )
D. ( frac{-1}{y^{n-1}}=2 sin x-frac{2}{1-n}+c e^{(n-1) sin x} x )
12
53 If ( y=left(sin ^{-1} xright)^{2}, ) then prove that ( (1- )
( left.x^{2}right) frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}=2 )
12
54 Find the solution of ( left(e^{y}+1right) cos x d x+ )
( e^{y} sin x d y=0 )
( A cdot sin xleft(e^{y}+1right)=c )
B cdot ( sin xleft(e^{y}-1right)=c )
( mathbf{c} cdot sin xleft(2 e^{y}+1right)=c )
D・sin ( xleft(3 e^{y}-1right)=c )
12
55 Solve:
( (mathbf{1}-boldsymbol{y}) boldsymbol{x} frac{d boldsymbol{y}}{d boldsymbol{x}}+(mathbf{1}+boldsymbol{x}) boldsymbol{y}=mathbf{0} )
12
56 The D. ( E ) of the family of parabolas having their focus at the origin and axis along the ( x ) -axis is
A ( cdot y_{1}left[y y_{1}-2 xright]=y )
B . ( y_{1}left(y_{1}right)^{2}=2 x y_{1}+y )
( mathbf{c} cdot y y_{1}^{2}+2 x y_{1}=y )
D. ( y y_{1}+2 x=y )
12
57 Solve:
( frac{d y}{d x}=e^{4 x-3 y} )
12
58 The differential equation of the family of
curves
( frac{boldsymbol{x}^{2}}{boldsymbol{a}^{2}}+frac{boldsymbol{y}^{2}}{boldsymbol{a}^{2}+boldsymbol{lambda}^{2}}=1 ) is ( (boldsymbol{lambda} ) is orbitary
constant)
A ( cdotleft(x^{2}-a^{2}right) y_{1}=x y )
B . ( left(x^{2}-a^{2}right) y_{2}-x y=0 )
C ( . x^{2} y_{2}-a^{2} y=0 )
D. ( left(x^{2}-a^{2}right) y_{1}+x y=0 )
12
59 The population of a city increases at the rate ( 3 % ) per year. If at time ( t ) the population of city is ( p, ) then find equation of p in time t.
A. ( _{p=c e} frac{3 t}{100} )
в. ( quad p=3 e^{frac{3 t}{100}} )
D. ( p=frac{3}{100} e^{3 t} )
12
60 Solve ( boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{2} boldsymbol{y}=boldsymbol{x}^{2} log boldsymbol{x} ) 12
61 I.F of ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{x} sin 2 boldsymbol{y}=boldsymbol{x}^{3} cos ^{2} boldsymbol{y} ) is:
A. ( tan y )
B. ( e^{tan y} )
( mathbf{c} cdot e^{sin } )
D. ( e^{x} )
12
62 Form the differential equation of the
family of curves represented by ( y^{2}= )
( (x-c)^{3} )
12
63 5.
A curve passes through the point 1,-
Let the slope of
sec
the curve at each point (x, y) be
/exo
(x
Then the equation of the curve is
(JEE Adv. 2013)
COs ec
e) sin() = log x + 6) cose()=logx +2
@sec ) 10g x +2 (a) cos(% )= log x +
12
64 The solution of ( frac{d y}{d x}=left(frac{y}{x}right)^{1 / 3} ) is:
( mathbf{A} cdot x^{2 / 3}+y^{2 / 3}=c )
B ( cdot y^{2 / 3}-x^{2 / 3}=c )
( mathbf{C} cdot x^{1 / 3}+y^{1 / 3}=c )
D ( cdot y^{1 / 3}-x^{1 / 3}=c )
12
65 The solution of ( y^{prime}-y=1, y(0)=1, ) is given by ( y(x)= )
( A cdot-exp (x) )
B. ( -exp (-x) )
( c cdot 1 )
D. ( 2 exp (x)-1 )
12
66 9.
4.
If y(x) satisfies the differential equation y’
= 2x secx and y(0)=0, then
erential equation y’ – ytanx
(2012)
@ 0526) v () = 1
3
32
12
67 Solve the differential equation ( frac{d y}{d x}+ )
( frac{y}{x}=x^{2} )
A. ( x=frac{x^{4}}{4}+C )
в. ( _{y}=frac{x^{4}}{4}+C )
c. ( _{x y}=frac{x^{4}}{4}+C )
D. None of these
12
68 Solve the differential equation: ( boldsymbol{y} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( frac{boldsymbol{x}}{boldsymbol{e}^{boldsymbol{y}}} )
12
69 If ( x^{y}=e^{x-y}, ) show that ( frac{d y}{d x}= ) ( frac{boldsymbol{y} log boldsymbol{x}}{boldsymbol{x}(log boldsymbol{x}+mathbf{1})} ) 12
70 The differential equation of all vertical lines in a plane is?
A ( cdot frac{d^{2} y}{d x^{2}}=0 )
B. ( frac{d^{2} x}{d y^{2}}=0 )
c. ( frac{d y}{d x}=0 )
D. ( frac{d x}{d y}=0 )
12
71 The differential equation satisfied by all
the straight lines ( x y ) – plane (not parallel to ( y ) -axis ) is:
A ( cdot frac{d y}{d x}=a ) constant
в. ( frac{d^{2} y}{d x^{2}}=0 )
c. ( _{y+frac{d y}{d x}}=0 )
D. ( frac{d^{2} y}{d x^{2}}+y=0 )
12
72 ntial equation
22.
Let y-y(x) be the solution of the differential ea
sin x dy + y cosx = 4x, x € (0,7). If y(0=0, then
COS X
dx
is equal to :
o
[JEE M 2018
12
73 Find the particular solution of the differential equation ( left(1+x^{2}right) frac{d y}{d x}= )
( left(e^{m tan ^{-1 x}}-yright), ) given that ( y=1 ) when ( x= )
0
12
74 Find the degree of each algebraic
expression
( 2 y^{2} z+10 y z )
12
75 Assertion
The differential equation of all straight
lines which are at a constant distance ( boldsymbol{p} )
from the origin is ( left(boldsymbol{y}-boldsymbol{x} boldsymbol{y}_{1}right)^{2}= )
( boldsymbol{p}^{2}left(mathbf{1}+boldsymbol{y}_{1}^{2}right) )
Reason
The general equation of any straight line which is at a constant distance ( boldsymbol{p} )
from the origin is ( boldsymbol{x} cos boldsymbol{alpha}+boldsymbol{y} sin boldsymbol{alpha}=boldsymbol{p} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
76 ff ( y=e^{a x}, ) then show that ( x frac{d y}{d x}=y log y ) 12
77 It is known that the cells of a given
bacterial culture divide every 3.5 hours (on average). If there are 500 cells in a dish to begin with, how many cells will there be after 12 hours?
A . 4653
в. 9876
( c .5383 )
D. None of these
12
78 Solve: ( boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}-boldsymbol{y}+boldsymbol{x} sin left(frac{boldsymbol{y}}{boldsymbol{x}}right)=mathbf{0} ) 12
79 The first order differential equation of
the family of circles with fixed radius ( mathbf{r} )
and with centre on ( x ) -axis is:
A ( cdot y^{2}left(frac{d y}{d x}right)^{2}+y^{2}=r^{2} )
B・ ( x^{2}left(frac{d y}{d x}right)^{2}+y^{2}=r^{2} )
c. ( left(frac{d y}{d x}right)^{2}+y^{2}=r^{2} )
D. ( y^{2}-left(frac{d y}{d x}right)^{2}=r^{2} )
12
80 Write degree of the differential equation ( frac{d^{2} y}{d x^{2}}+xleft(frac{d y}{d x}right)^{2}=2 x^{2} log left(frac{d^{2} y}{d x^{2}}right) ) 12
81 The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in ( 2000, ) what will be
the population in ( 2010 ? )
12
82 For the following differential equation, find the general solution.
( frac{d y}{d x}+x=1 )
12
83 Solve:
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}+boldsymbol{y}+mathbf{1}}{boldsymbol{x}+boldsymbol{y}-mathbf{1}} ) when ( boldsymbol{y}=frac{mathbf{1}}{mathbf{3}} ) at ( boldsymbol{x}=frac{mathbf{2}}{mathbf{3}} )
12
84 Write the differential equation
representing the family of curves ( boldsymbol{y}= )
( m x, ) where ( m ) is an arbitrary constant.
12
85 The Integrating factor of the differential equation ( left(1-y^{2}right) frac{d x}{d y}+y x=a y ) is
A ( cdot frac{1}{y^{2}-1} )
B. ( frac{1}{sqrt{y^{2}-1}} )
c. ( frac{1}{1-y^{2}} )
D. ( frac{1}{sqrt{1-y^{2}}} )
12
86 dy
– ty = 1 and y(0)=-1, then
3.
dt
If y(t) is a solution of (1+t)
It
y(1) is equal to
(20035
(a) – 1/2
(c) e-1/2
(b) e+1/2
(d) 1/2
12
87 The rate of change of volume of sphere with respect to its surface area ( S ) is
A. ( sqrt{frac{S}{pi}} )
в. ( frac{1}{2} sqrt{frac{S}{pi}} )
( ^{c} cdot frac{1}{4} sqrt{frac{S}{pi}} )
D. ( 4 sqrt{frac{S}{pi}} )
12
88 The D.E of the family of parabolas with vertex at (0,-1) and having axis along the y axis is
A. ( x y^{1}-2 y-2=0 )
В. ( x y^{1}+y+1=0 )
c. ( x y^{1}-y-1=0 )
D. ( x y^{1}+2 x y+1=0 )
12
89 The differential equation obtained by eliminating arbitrary constants from ( boldsymbol{y}=boldsymbol{a} cdot boldsymbol{e}^{b boldsymbol{x}}, ) is
A ( cdot y frac{d^{2} y}{d x^{2}}+frac{d y}{d x}=0 )
B. ( y frac{d^{2} y}{d x^{2}}-frac{d y}{d x}=0 )
c. ( y frac{d^{2} y}{d x^{2}}-left(frac{d y}{d x}right)^{2}=0 )
D. ( y frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}=0 )
12
90 Solve the following differential equation ( left(x^{2}+1right) frac{d y}{d x}=1 ) 12
91 Find the sum of the order and degree of
the differential equation ( boldsymbol{y}= )
( boldsymbol{x}left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{3}+frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} )
12
92 Solve the differential equation:
( sec ^{2} y frac{d y}{d x}+tan y=x^{3} )
A ( cdot tan y=x^{3}+3 x^{2}+6 x-6+c e^{-x} )
B . ( tan y=x^{3}-3 x^{2}-6 x-6+c e^{-x} )
C. ( tan y=x^{3}-3 x^{2}+6 x-6+c e^{-x} )
D. None of these.
12
93 The integrating factor of the differential equation ( 3 x log _{e} x frac{d y}{d x}+y=2 log _{e} x ) is
given by:
A ( cdotleft(log _{e} xright)^{2} )
B. ( log _{e}left(log _{e} xright) )
( mathbf{c} cdot log _{e} x )
D. ( quadleft(log _{e} xright)^{frac{1}{3}} )
12
94 Solve the differential equation ( left(x^{2}-right. )
( left.boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{x}+2 boldsymbol{x} boldsymbol{y} boldsymbol{d} boldsymbol{y}=0 ; ) given that ( boldsymbol{y}=mathbf{1} )
when ( boldsymbol{x}=mathbf{1} )
12
95 Consider a differential equation of order
( m ) and degree ( n . ) Which one of the following pairs is not feasible?
A ( cdot(3,2) )
В ( cdotleft(2, frac{3}{2}right) )
c. (2,4)
(年. ( (2,4)) )
D. (2,2)
12
96 The general solution of the differential
equation ( (x+y) d x+x d y=0 ) is
A ( cdot x^{2}+y^{2}=C )
B . ( 2 x^{2}-y^{2}=C )
c. ( x^{2}+2 x y=C )
D. ( y^{2}+2 x y=C )
12
97 Find the degree of the differential equation: ( sqrt{1+x^{2}}=frac{d y}{d x} )
( A cdot 2 )
B.
( c cdot 4 )
D. None of these
12
98 The solution of ( frac{d y}{d x}=frac{y^{2}}{x y-x^{2}} )
A. ( y=c e^{x y} )
B. ( y=frac{e^{frac{y}{x}}}{c} )
c. ( log y=x y+c )
D. ( log x=x y+c )
12
99 Let ( Gamma ) denote a curve ( y=f(x) ) which is
in the first quadrant and let the point
(1,0) lie on it. Let the tangent to ( Gamma ) at a
point ( boldsymbol{P} ) intersect the ( mathbf{y} ) -axis at ( boldsymbol{Y}_{boldsymbol{P}} . ) If
( P Y_{P} ) has length 1 for each point ( P ) on ( Gamma )
Then which of the following options
is/are correct?
This question has multiple correct options
A ( y=-ln left(frac{1+sqrt{1-x^{2}}}{x}right)+sqrt{1-x^{2}} )
B. ( x y^{prime}+sqrt{1-x^{2}}=0 )
C ( cdot x y^{prime}-sqrt{1-x^{2}}=0 )
D. ( y=ln left(frac{1+sqrt{1-x^{2}}}{x}right)-sqrt{1-x^{2}} )
12
100 The differential equation of all circles passing through the origin and having their centers on the x-axis is:
A ( cdot y^{2}=x^{2}+2 x y frac{d y}{d x} )
B. ( y^{2}=x^{2}-2 x y frac{d y}{d x} )
c. ( x^{2}=y^{2}+2 x y frac{d y}{d x} )
D. None of these
12
101 Solution of ( boldsymbol{y}^{2} boldsymbol{d} boldsymbol{x}+left(boldsymbol{x}^{2}-boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{y}= )
( mathbf{0} )
A ( cdot tan ^{-1}left(frac{x}{y}right)+log y+c=0 )
B ( cdot 2 tan ^{-1}left(frac{x}{y}right)+log y+c=0 )
( mathbf{c} cdot log y(y+sqrt{x^{2}+y^{2}})+log y+c=0 )
D cdot ( log y(y-sqrt{x^{2}+y^{2}})+log y+c=0 )
12
102 The differential equation of all
parabolas with axis parallel to the axis
of ( boldsymbol{y} ) is:
A ( cdot y_{2}=2 y_{1} )
в. ( y_{3}=2 y_{1} )
c. ( y_{2}^{3}=y_{1} )
D. none of these
12
103 The D.E whose solution is ( x y=a x^{2}+frac{b}{x} )
is
A ( cdot x^{2} y_{2}+2 x y_{1}=2 y )
в. ( x^{2} y_{2}-x y_{1}+2 y=0 )
c. ( x^{2} y_{2}+x y_{1}+y=0 )
D . ( x^{2} y_{2}+x y_{1}+2 y=0 )
12
104 If the differential equation representing the family of all circles touching ( x- ) axis at the origin is ( left(x^{2}-y^{2}right) frac{d y}{d x}= )
( boldsymbol{g}(boldsymbol{x}) boldsymbol{y}, ) then ( boldsymbol{g}(boldsymbol{x}) ) equals:
A ( cdot frac{1}{2} x )
в. ( 2 x^{2} )
c. ( 2 x )
D. ( frac{1}{2} x^{2} )
12
105 Show that ( y=frac{1}{x} ) is a solution of the differential equation ( frac{d y}{d x}=log x ) 12
106 The solution of ( frac{d y}{d x}+y tan x=cos ^{2} x ) is:
A ( cdot y sec ^{2} x=c+sin x )
B. ( y sec x=c+cos x )
c. ( y sec ^{2} x=c+cos x )
D. ( y sec x=c+sin x )
12
107 Show that the family of curves for which ( frac{d y}{d x}=frac{x^{2}+y^{2}}{2 x y}, ) is given by ( x^{2}-y^{2}= )
( C X . ) differential equation is :
12
108 The number of people having flu after 20 days is
A. 1400
B. 1540
( c .1498 )
D. 1492
12
109 The solution of the differential equation
( x d x+y d y=0 ) is ( f(x, y)=c ) passes
through (1,1) then value of ( c ) is :
( A cdot-2 )
B. 2
( c .-1 )
D.
12
110 The degree of the differential equation ( boldsymbol{y}_{3}^{2 / 3}+mathbf{2}+mathbf{3} boldsymbol{y}_{2}+boldsymbol{y}_{1}=mathbf{0} ) is:
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D. none of these
12
111 If ( mathbf{Y}=(x+sqrt{x^{2}-1})^{m} ) show that
( left(x^{2}-1right) frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}=m^{2} y )
12
112 The differential equation, which represents the family if plane curves
( boldsymbol{y}=boldsymbol{e}^{c boldsymbol{x}}, ) is
A ( cdot y^{prime}=c y )
B. ( x y^{prime}-log y=0 )
c. ( x log y=y y^{prime} )
D. ylogy = ( x y^{prime} )
12
113 Represent the following families of curves by forming the corresponding differential equation(a, b being parameters). ( x^{2}+y^{2}=a^{2} ) 12
114 The differential equation ( frac{d y}{d x}= )
( sqrt{1-y^{2}} ) determines a family of circles ( boldsymbol{y} )
with:
A . variable radii and a fixed centre (0,1)
B. variable radii and a fixed centre (0,-1)
C. fixed radius 1 and variable centres along the ( x ) -axis
D. fixed radius 1 and variable centres along the ( y ) -axis
12
115 D.E of the parabolas having x-axis as
axis and origin as focus is:
( mathbf{A} cdot yleft(frac{d y}{d x}right)^{2}+4 x frac{d y}{d x}=4 )
B. ( 2 x frac{d y}{d x}-y=0 )
( ^{mathbf{C}} yleft(frac{d y}{d x}right)^{2}+y=2 x y frac{d y}{d x} )
( yleft(frac{d y}{d x}right)^{2}+2 x frac{d y}{d x}-y=0 )
12
116 If the solution of the differential
equation ( x frac{d y}{d x}+y=x e^{x} ) be, ( x y= )
( e^{x} varphi(x)+c ) then ( varphi(x) ) is equal to:
A. ( x+1 )
B. ( x-1 )
c. ( 1-x )
D.
12
117 Solution of differential equation
( sin y cdot frac{d y}{d x}+frac{1}{x} cos y=x^{4} cos ^{2} y ) is
A ( cdot x ) sec ( y=x^{6}+C )
B. ( 6 x ) sec ( y=x+C )
c. ( 6 x ) secy ( =x^{6}+C )
D. ( 6 x ) sec ( y=6 x^{6}+C )
12
118 The solution of ( frac{d y}{d x}+1=e^{x+y} ) is:
A ( cdot e^{-(x+y)}+x+c=0 )
B ( cdot e^{-(x+y)}-x+c=0 )
c. ( e^{x+y}+x+c=0 )
D. ( e^{x+y}-x+c=0 )
12
119 A solution of the differential equation ( left(frac{d y}{d x}right)-x frac{d y}{d x}+y=0 ) is 12
120 The order and the degrees of the differential equation of all ellipses with centre at the origin, major axis along ( x- )
axis and eccentricity ( frac{sqrt{mathbf{3}}}{2} ) are
respectively:
A . 1,1
в. 2,1
c. 1,2
D. 2,
12
121 The solution of differential equation ( 2 x frac{d y}{d x}-y=3 ) represents
A. a straight line
B. an ellipse
c. a parabola
D. circles
12
122 The order and degree of the differential equation ( sqrt{frac{boldsymbol{d y}}{boldsymbol{d} boldsymbol{x}}}-4 frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}-boldsymbol{7} boldsymbol{x}=boldsymbol{0} ) are 12
123 The package’s velocity at the instant the parachute opens is (approximately nearest whole number)
A. ( 84 mathrm{m} / mathrm{s} )
B. ( 82 mathrm{m} / mathrm{s} )
( c cdot 89 mathrm{m} / mathrm{s} )
D. ( 92 mathrm{m} / mathrm{s} )
12
124 ( frac{e^{-2 sqrt{x}}-y}{sqrt{x}} frac{d x}{d y}=1 ) 12
125 Find the solution of ( frac{d y}{d x}=frac{2 y-x-4}{y-3 x+3} )
( mathbf{A} cdotleft(log left(v^{2}-3 v+1right)-frac{1}{2 frac{sqrt{21}}{2}} log frac{v-frac{5}{2}-frac{sqrt{21}}{2}}{v-frac{5}{2}+frac{sqrt{21}}{2}}right)= )
( -log alpha+c )
( ^{mathrm{B}} cdotleft(log left(v^{2}+3 v+1right)-frac{1}{2 frac{sqrt{21}}{2}} log frac{3 v-frac{5}{2}-frac{sqrt{21}}{2}}{v-frac{5}{2}+frac{sqrt{21}}{2}}right)= )
( -log alpha+c )
( left(log left(v^{2}-5 v+1right)-frac{1}{2 frac{sqrt{21}}{2}} log frac{3 v-frac{5}{2}-frac{sqrt{21}}{2}}{v-frac{5}{2}+frac{sqrt{21}}{2}}right)= )
( -log alpha+c )
( left(log left(v^{2}-5 v+1right)-frac{1}{2 frac{sqrt{21}}{2}} log frac{v-frac{5}{2}-frac{sqrt{21}}{2}}{v-frac{5}{2}+frac{sqrt{21}}{2}}right)= )
( -log alpha+c )
12
126 The D.E whose solution is ( y=A cos x+ )
( sin x ) is:
A ( cdot frac{d y}{d x}+y tan x=sec x )
в. ( frac{d y}{d x}+y sin x=cos x )
c. ( frac{d y}{d x}+y sec x=tan x )
D. ( frac{d y}{d x}+y cot x=csc x )
12
127 D.E. whose solution is ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 )
A ( . x y y_{2}+x y_{1}^{2}=y y_{1} )
В. ( x y y_{2}+y_{1}=y )
C ( . x^{2} y_{2}+x y_{1}=y )
D. ( x^{2} y_{2}+2 x y_{1}=y_{1}^{2} )
12
128 The solution of primitive integral equation (x2 + y4) dy=xy
dx is y= y(x). Ify (1) = 1 and (x)= e, then xo is equal to
(2005)
(a) 262-1)
(b) 126e? +1)
e2 +1
(c)
e
(d)
V
2
12
129 Solve the given differential equation. ( frac{d y}{d x}+frac{x sqrt{left(x^{2}+y^{2}right)}-y^{2}}{x y}=0 )
A ( cdot sqrt{left(x^{2}+y^{2}right)}=x log k / x )
B . ( sqrt{left(x^{2}+y^{2}right)}=log k / x )
C ( cdot sqrt{left(x^{2}+y^{2}right)}=x^{2} log k / x )
D. none of these
12
130 Find a particular solution of the differential equation ( (x+1) frac{d y}{d x}= )
( 2 e^{-y}-1, ) given that ( y=0 ) when ( x=0 )
12
131 Solve: ( cos ^{2} x frac{d y}{d x}+y=tan x ) 12
132 The degree of ( frac{d^{2} y}{d x^{2}}+ )
( left(1+left(frac{d y}{d x}right)right)^{3 / 2}=0 )
( A cdot 2 )
B.
( c cdot 4 )
D. 6
12
133 Find solution for ( D E Rightarrowleft(frac{d y}{d x}right)^{2} ) ( boldsymbol{y} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{x}=mathbf{0} ) 12
134 The order and degree of the differential equation
[2002]
(a) (1,5)
(C) (3,3)
(b) (3,1)
(d) (1,2)
12
135 If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sin boldsymbol{x}, ) find ( boldsymbol{f}^{prime}(boldsymbol{pi}), ) using first
principle.
12
136 Find the differential equation of ( y=A ) ( operatorname{cox}+B sin x ) 12
137 Let ( _{y}-y(x) ) be the solution of the
differential equation ( sin x frac{d y}{d x}-y cos x- )
( 4 x,_{x} epsilon(0, pi) . i f yleft(frac{pi}{2}right)=0 )
then ( yleft(frac{pi}{6}right) ) is equal to
A ( cdot-frac{4}{9} pi^{2} )
B. ( -frac{4}{9 sqrt{3}} pi^{2} )
c. ( -frac{-8}{9 sqrt{3}} pi^{2} )
D. None of these
12
138 Solution of the differential equation ( left(x^{2}+y^{3}right)left(2 x^{2} d x+3 y d yright)=12 x d x+ )
( 18 y^{2} d y ) is
A ( cdot frac{2}{3} x^{3}+frac{3}{2} y^{2}=6 ln left(x^{2}+y^{3}right)+c )
в. ( x^{2}+y^{3}=9 ln left(x^{2}+y^{3}right)+c )
c. ( frac{2}{3} x^{3}+frac{3}{2} y^{2}=6 ln left(x^{3}+y^{2}right)+c )
D. ( x^{3}+y^{2}=6 ln left(x^{2}+y^{3}right)+c )
12
139 Solve the following differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}^{2}+boldsymbol{x}-frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x} neq mathbf{0} ) 12
140 If ( boldsymbol{y}(boldsymbol{t}) ) is a solution of ( (boldsymbol{1}+boldsymbol{t}) frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{t}}-boldsymbol{t} boldsymbol{y}= )
1 and ( y(0)=-1, ) then ( y(1) ) equal to
A. ( -frac{1}{2} )
B. ( e+frac{1}{2} )
c. ( _{e-frac{1}{2}} )
D. ( frac{1}{2} )
12
141 Integrating factor of ( frac{boldsymbol{x} boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}-boldsymbol{y}=boldsymbol{x}^{4}-boldsymbol{3} boldsymbol{x} )
is:
12
142 Consider the following statements:
1. The general solution of ( frac{d y}{d x}=f(x)+ ) ( x ) is of the form ( y=g(x)+c, ) where ( c ) is
an arbitrary constant.
2. The degree of ( left(frac{d y}{d x}right)^{2}=f(x) ) is 2 Which of the above statements is/are
correct?
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
143 ( sqrt{1-x^{2}}+sqrt{1-y^{2}}=a(x-y) )
Then prove that ( frac{d y}{d x}=frac{sqrt{1-y^{2}}}{sqrt{1-x^{2}}} )
12
144 If ( boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{y}(log boldsymbol{y}-log boldsymbol{x}+1), ) then the
solution of the equation is?
12
145 Form the differential equation corresponding to ( boldsymbol{x} boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+frac{boldsymbol{b}}{boldsymbol{x}},(boldsymbol{a}, boldsymbol{b}) ) 12
146 Solve the differential equation ( frac{d y}{d x}+ )
( boldsymbol{y}=boldsymbol{e}^{-boldsymbol{x}} )
12
147 Solve the following differential equation ( frac{d y}{d x}+1=sin x ) 12
148 The curve satisfying ( frac{mathbf{d}^{2} boldsymbol{y}}{mathbf{d} boldsymbol{x}^{2}}-mathbf{4} boldsymbol{y}^{prime}=mathbf{0} ) and
passing through (1,0) is:
A ( cdot y=x e^{4 x} )
B . ( y=a e^{4 x} )
c. ( y=e^{4 x}-e^{4} )
D・ ( y=aleft(e^{4 x}-e^{4}right) )
12
149 The differential equation representing
the family of curves ( y=x e^{c x} ) (c is a
constant) is
A ( cdot frac{d y}{d x}=frac{y}{x}left(1-log frac{y}{x}right) )
B . ( frac{d y}{d x}=frac{y}{x} log left(frac{y}{x}right)+1 )
C ( cdot frac{d y}{d x}=frac{y}{x}left(1+log frac{y}{x}right) )
D. ( frac{d y}{d x}+1=frac{y}{x} log frac{y}{x} )
12
150 The solution of the differential equation
( boldsymbol{y}^{prime}=frac{mathbf{1}}{e^{-boldsymbol{y}}-boldsymbol{x}}, ) is
( mathbf{A} cdot x=e^{-y}(y+c) )
B . ( y+e^{-y}=x+c )
C ( . x=e^{y}(y+c) )
D. ( x+y=e^{-y}+c )
12
151 Determine the differential equation of
parabolas with foci at origin and axes
along X-axis.
Hint: ( y^{2}=4 a(x-a) )
( mathbf{A} cdot y^{3}left(frac{d y}{d x}right)^{2}=2 x y frac{d y}{d x}-1 )
B ( y^{4}left(frac{d y}{d x}right)^{2}=2 x y frac{d y}{d x}-1 )
( ^{mathbf{C}} y^{3}left(frac{d y}{d x}right)^{2}=2 x y frac{d y}{d x}-2 )
D ( y^{4}left(frac{d y}{d x}right)^{3}=2 x y frac{d y}{d x}-1 )
12
152 Show that ( 3 e^{x} tan y d x+(1- )
( left.e^{x}right) sec ^{2} y d y=0 )
12
153 If ( sqrt{1-x^{2}}+sqrt{1-y^{2}}=a(x-y) ) then
prove that ( frac{d y}{d x}=sqrt{frac{1-y^{2}}{1-x^{2}}} )
12
154 Find the solution of the differential
equation:
( left(1+y^{2}right)+left(x-e^{t a n^{-1} y}right) frac{d y}{d x}=0 )
12
155 Solve the differential equation:
( left(x^{2}-1right) frac{d y}{d x}+2(x+2) y=2(x+1) )
( frac{y(x+1)^{3}}{x+1}=left{frac{(x+1)^{2}}{2}-4(x+1)+4 log (x+1)right}+c )
( frac{y(x-1)^{3}}{x+1}=left{frac{(x+1)^{2}}{2}-4(x+1)+4 log (x+1)right}+c )
( frac{y(x-1)^{3}}{x-1}=left{frac{(x+1)^{2}}{2}-4(x+1)+4 log (x+1)right}+c )
D. None of these
12
156 Solve the differential equation ( left[frac{e^{-2 sqrt{x}}}{sqrt{x}}-frac{y}{sqrt{x}}right] frac{d x}{d y}=1 quad dots(x neq 0) ) 12
157 The order and degree of the differentia equation ( y=frac{d y}{d x} x+sqrt{a^{2}left(frac{d y}{d x}right)^{2}+b^{2}} )
is
( A cdot 3,1 )
в. 1,3
( c .2,1 )
D. 1,2
12
158 Find all functions ( f(x) ) defined on ( left(-frac{pi}{2}, frac{pi}{2}right) ) with real values and has a
primitive ( F(x) ) such that ( f(x)+ ) ( cos x cdot F(x)=frac{sin 2 x}{(1+sin x)^{2}} . ) Then find
( f(x) )
A ( cdot f(x)=-frac{2 cos x}{(1+sin x)^{2}}-C e^{-cos x} cdot sin x )
B. ( f(x)=-frac{2 cos x}{(1+sin x)^{2}}-C e^{-sin x} cdot cos x )
C ( quad f(x)=-frac{2 cos x}{(1+cos x)^{2}}-C e^{sin x} cdot sin x )
D. ( f(x)=-frac{2 cos x}{(1+sin x)^{2}}-C e^{cos x} cdot cos x )
12
159 7.
For the primitive integral equation ydx + yżdy = x dy;
XER, y>0, y=y(x), y(1) = 1, then y(-3) is (2005S)
(a) 3 (6) 2 (c) 1 0 (d) 5
12
160 The differential equation of all
parabolas with axis parallel to the axis
of ( boldsymbol{y} ) is:
A ( cdot y_{2}=2 y_{1} )
в. ( y_{3}=2 y_{1} )
c. ( y_{2}^{3}=y_{1} )
D. none of these
12
161 Solve ( frac{d y}{d x}=cos (x+y) )
Solve ( frac{d y}{d x}=cos (x+y) )
12
162 Solve:
( frac{cos ^{2} y}{x} d y+frac{cos ^{2} x d x}{y}=0 )
12
163 Solve :
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=left(boldsymbol{1}+boldsymbol{x}^{2}right)left(boldsymbol{1}+boldsymbol{y}^{2}right) )
12
164 If ( y=a cos (log x)-b sin (log x), ) then
the value of ( x^{2} frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}+y ) is
( A cdot 0 )
B.
( c cdot 2 )
D.
12
165 The solution of ( e^{y}left(1+x^{2}right) frac{d y}{d x}=2 x(1+ )
( left.e^{y}right) ) is:
A ( cdot frac{1+e^{y}}{1+x^{2}}=c )
B ( cdot e^{y}left(1+x^{2}right)=c )
C ( cdotleft(1+e^{y}right)+left(1+x^{2}right)=c )
D. ( left(e^{y}+1right) x^{2}=c )
12
166 1.
2.
The differential equation representing the family of curves
12
where c is a positive parameter, is of
(1999 – 3 Marks)
(a) order 1 (b) order 2 (c) degree 3 (d) degree 4
12
167 The equation of motion of a body falling under gravity is given by ( frac{d v}{d t}=g- ) ( frac{g}{lambda^{2}} v^{2} . ) The Velocity as a function of time
is? given that ( boldsymbol{v}=mathbf{0} boldsymbol{t}=mathbf{0} )
A ( cdot v=lambda^{2} tan h^{-1} frac{g t}{lambda} )
B. ( v=lambda cot h^{-1} frac{g t}{lambda} )
c. ( v=lambda tan h^{-1} frac{g t}{lambda} )
D. ( v=lambda cot ^{2} h^{-1} frac{g t}{lambda} )
12
168 Form the differential equation
corresponding to ( boldsymbol{y}=boldsymbol{a} cos (boldsymbol{n} boldsymbol{x}+ )
( boldsymbol{b}),(boldsymbol{a}, boldsymbol{b}) )
12
169 Find the I.F of ( boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{y}(boldsymbol{1}+boldsymbol{x})=mathbf{1} )
A . ( x . e^{x} )
в. ( e^{x} / x )
c. ( x+log x )
D. xlogx
12
170 Find differential equation of all circles in the first quadrant which touch the co-ordinate axis. 12
171 f ( y=sin left(2 sin ^{-1} xright), ) then prove that
( frac{d y}{d x}=2 sqrt{frac{1-y^{2}}{1-x^{2}}} )
12
172 The population of a certain country is known to increase at a rate proportional to the number of people presently living
in the country. If after two years the population has doubled, and after three years the population is 20000 , estimate the number of people initially living in
the country.
A .607
в. 707
c. 7061
D. 6077
12
173 20
15. Let I be the purchase value of an equipment and V (t) be the
value after it has been used for t years. The value V(t)
depreciates at a rate given by differential equation
dV (t) = -k(17
-=-k(T -t), where k> 0 is a constant and T is the
dt
total life in years of the equipment. Then the scrap value
V(T) of the equipment is
[2011]
(a) 1-k!
(b) 1- k(T – 1)2
2
(c) e-kł
(d) T² – 1
12
174 Solve ( left(boldsymbol{y}+mathbf{3} boldsymbol{x}^{2}right) frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{d} boldsymbol{y}}=boldsymbol{x} ) 12
175 Equation of the curve through the origin satisfying ( boldsymbol{d} boldsymbol{y}=(sec boldsymbol{x}+boldsymbol{y} tan boldsymbol{x}) boldsymbol{d} boldsymbol{x} ) is:
A. ( y sin x=x )
B. ( y cos x=x )
( mathbf{c} cdot y tan x=x )
D. none of these
12
176 Solve ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+frac{boldsymbol{2} boldsymbol{y}}{boldsymbol{x}}=boldsymbol{e}^{boldsymbol{x}} ) 12
177 ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}+boldsymbol{y}+mathbf{1}}{mathbf{2} boldsymbol{x}+mathbf{2} boldsymbol{y}+mathbf{3}} ) 12
178 The solution of ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x} boldsymbol{y}}{boldsymbol{x}^{2}+boldsymbol{y}^{2}} ) is
A. ( quad frac{x^{2}}{y^{2}} )
[
x=c e^{frac{1}{2}}
]
в. ( quad y=c e^{frac{x^{2}}{y^{2}}} )
c.
[
y=c e^{frac{x^{2}}{2 y^{2}}}
]
D. ( quad x=c e^{frac{2 x^{2}}{y^{2}}} )
12
179 Find the particular solution of the differential equation:
( boldsymbol{y}(mathbf{1}+log boldsymbol{x}) frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{d} boldsymbol{y}}-boldsymbol{x} log boldsymbol{x}=mathbf{0} )
when ( y=e^{2} ) and ( x=e )
12
180 Find ( frac{d y}{d x} )
( left(x^{3}-2 y^{3}right) d x+3 x y^{2} d y=0 )
12
181 The slope of a curve at each of its points is equal to the square of the abscissae of the point. Find the particular curve
through the point (-1,1) ( mathbf{1}+boldsymbol{y}=boldsymbol{2} boldsymbol{e}^{boldsymbol{x}^{2} / mathbf{2}} )
12
182 The differential equation whose general
solution is ( boldsymbol{y}=boldsymbol{A} cos (boldsymbol{x}+boldsymbol{3}), ) where ( boldsymbol{A} )
is arbitrary constant is
( mathbf{A} cdot cot (x+3) y_{1}+y=0 )
B cdot ( tan (x+3) y_{1}+y=0 )
( mathbf{c} cdot cot (x+3) y_{1}-y=0 )
D cdot ( tan (x+3) y_{1}-y=0 )
12
183 Order and degree of ( left(1+y_{1}^{2}right) y_{3}= )
( mathbf{3} boldsymbol{y}_{2} boldsymbol{y}_{1}^{2} ) are:
A . 2,3
B. 2,
( c cdot 3, )
D. 3,2
12
184 If ( y=e^{m sin ^{-1} x} ) then show that ( (1- )
( left.boldsymbol{x}^{2}right) boldsymbol{y}_{boldsymbol{n}+mathbf{2}}-(boldsymbol{2} boldsymbol{n}+mathbf{1}) boldsymbol{x} boldsymbol{y}_{boldsymbol{n}+mathbf{1}}-left(boldsymbol{n}^{2}+right. )
( left.boldsymbol{m}^{2}right) boldsymbol{y}_{n}=mathbf{0} )
12
185 The soultion of ( 3 e^{x} cos ^{2} y d x+(1- )
( left.e^{x}right) cot y d y=0 ) is
12
186 Determine the order and degree(if defined) of the following differentia
equation. ( left(frac{boldsymbol{d} boldsymbol{s}}{boldsymbol{d} boldsymbol{t}}right)^{boldsymbol{4}}+boldsymbol{3} boldsymbol{s} frac{boldsymbol{d}^{2} boldsymbol{s}}{boldsymbol{d} boldsymbol{t}^{2}}=mathbf{0} )
12
187 Solve the differential equation:
( (x-sqrt{x y}) d y=y d x )
12
188 Solve the given differential equation:
( boldsymbol{y}^{2} boldsymbol{d} boldsymbol{x}+left(boldsymbol{x}^{2}-boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{y}=mathbf{0} )
12
189 Form the differential equation of the family of curves represented by the equation(a being the parameter). ( (x-a)^{2}+2 y^{2}=a^{2} ) 12
190 If ( y_{1}(x) ) is a solution of the differential equation ( frac{d y}{d x}+f(x) y=0, ) then ( a ) solution of differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+ )
( boldsymbol{f}(boldsymbol{x}) boldsymbol{y}=boldsymbol{r}(boldsymbol{x}) ) is
A ( cdot frac{1}{y(x)} int y_{1}(x) r(x) d x )
B. ( y_{1}(x) int frac{r(x)}{y_{1}(x)} d x )
c. ( f r(x) y_{1}(x) d x )
D. None of these
12
191 The order of differential equation of all
circles of given radius ( ^{prime} a^{prime} ) is
A . 4
B. 2
( c cdot 1 )
D. 3
12
192 Solve ( sec ^{2} x tan y d y+sec ^{2} y tan x d x= )
( mathbf{D} )
12
193 What is the solution of ( (1+2 x) d y- )
( (1-2 y) d x=0 ? )
A. ( x-y-2 x y=c )
в. ( y-x-2 x y=c )
c. ( y+x-2 x y=c )
D. ( x+y+2 x y=c )
12
194 The D.E of simple harmonic motion
whose solution is given by ( boldsymbol{x}= ) ( A cos (n t+alpha) ) is
A ( cdot frac{d^{2} x}{d t^{2}}+n x=0 )
B. ( frac{d^{2} x}{d t^{2}}+n^{2} x=0 )
c. ( frac{d^{2} x}{d t^{2}}-n^{2} x=0 )
D. ( frac{d^{2} x}{d t^{2}}+frac{1}{n^{2}} x=0 )
12
195 The differential equation of the family of lines which pass through (1,-1) is:
A ( cdot y=(x+1) frac{d y}{d x}+1 )
B. ( y=(x+1) frac{d y}{d x}-1 )
c. ( y=(x-1) frac{d y}{d x}+1 )
D. ( y=(x-1) frac{d y}{d x}-1 )
12
196 Prove that
( y=2 e^{x}+3 e^{2} x ) is not g.s. of differential
equation ( boldsymbol{y}^{2}-mathbf{3} boldsymbol{y}+mathbf{2} boldsymbol{y}=mathbf{0} )
12
197 What is the degree of the differential quation ( left(frac{d^{3} y}{d x^{3}}right)^{3 / 2}=left(frac{d^{2} y}{d x^{2}}right)^{2} ? )
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 3 )
( D )
12
198 An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given
quantity of water. Show that the cost of the material will be least when depth of the tank is half of its width.
12
199 Solve the following differential
equations:
( boldsymbol{x} boldsymbol{d} boldsymbol{x}+boldsymbol{y} boldsymbol{d} boldsymbol{y}=boldsymbol{x} boldsymbol{d} boldsymbol{y}-boldsymbol{y} boldsymbol{d} boldsymbol{x} )
A ( cdot ln left(x^{2}+y^{2}right)=2 tan ^{-1}left(frac{y}{x}right)+c )
B cdot ( ln (x+y)^{2}=2 tan ^{-1}left(frac{y}{x}right)+c )
c. ( ln left(x^{2}+y^{2}right)=2 tan ^{-1}left(frac{x}{y}right)+c )
D ( cdot ln (x+y)^{2}=2 tan ^{-1}left(frac{x}{y}right)+c )
12
200 Eliminate the arbitrary constant and obtain the differential equation satisfied by it
( boldsymbol{y}=boldsymbol{2} boldsymbol{x}+boldsymbol{c} boldsymbol{e}^{boldsymbol{x}} )
A ( cdot y^{prime}-y=2(1-x) )
В. ( y^{prime}-y=2(3-x) )
c. ( y^{prime}-y=2(1-3 x) )
D . ( y^{prime}-y=3(1-x) )
12
201 The general solution of the differential
equation ( y d x-x d y+x^{2} cdot sin y d y+ )
( left(1+x^{2}right) d x=0, ) is equal to:
A ( cdot x^{2}=(c+1) x+x cdot cos y-y )
B . ( x^{2}=(c+1) x+x . cos y+y )
c. ( x^{2}=c x+x . cos y+y-1 )
D. ( x^{2}=c x+x . cos y+y+1 )
12
202 Solve the differential equation:
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}-boldsymbol{y} tan boldsymbol{x}=mathbf{0} )
12
203 Obtain the differential equation from
the relation ( A x^{2}+B y^{2}=1, ) where ( A )
and B are constants.
12
204 The differential equation for the family of curves ( x^{2}+y^{2}-2 a y=0, ) where ( a ) is
an arbitrary constant is
A ( cdot 2left(x^{2}-y^{2}right) y^{prime}=x y )
B ( cdot 2left(x^{2}+y^{2}right) y^{prime}=x y )
c. ( left(x^{2}-y^{2}right) y^{prime}=2 x y )
D. ( left(x^{2}+y^{2}right) y^{prime}=2 x y )
12
205 Find the particular solution of the differential equation
( left(1-y^{2}right)(1+log x) d x+2 x y d y=0 )
given that ( y=0 ) when ( x=1 )
12
206 The order and degree of the differential
equation. ( left(frac{d^{2} y}{d x^{2}}right)^{3}+left(frac{d y}{d x}right)=int y d x ) are
respectively.
A . 2 and 3
B. 2 and 2
( c .3 ) and 1
D. 3 and 2
12
207 If ( y=sqrt{x+sqrt{x+sqrt{x}}}+ldots . infty, ) then
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? )
A. ( frac{1}{y^{2}-1} )
в. ( frac{1}{2 y+1} )
c. ( frac{2 y}{y^{2}-1} )
D. ( frac{1}{2 y-1} )
12
208 The solution of differential equation
( boldsymbol{x}^{2} boldsymbol{y}^{2} boldsymbol{d} boldsymbol{y}=left(1-boldsymbol{x} boldsymbol{y}^{3}right) d boldsymbol{x} ) is
A ( cdot x^{3} y^{3}=x^{2}+C )
B. ( 2 x^{3} y^{3}=3 x^{2}+C )
c. ( x^{3} y^{3}=x^{2}+x+C )
D. ( x^{3} y^{3}=3 x^{2}+C )
12
209 The D.E whose solution is ( y=c(x-c)^{2} )
is
( ^{mathbf{A}} cdotleft(frac{d y}{d x}right)^{3}=4 yleft(x frac{d y}{d x}-2 yright) )
В. ( y_{1}^{3}=2 yleft(x y_{1}-yright) )
( ^{mathbf{c}}left(frac{d y}{d x}right)^{3}=4 yleft(2 x frac{d y}{d x}-yright) )
D. ( left(frac{d y}{d x}right)^{3}=2 yleft(x frac{d y}{d x}-4 yright) )
12
210 Form the differential equation by eliminating the arbitrary constants from the equation ( y=a cos (2 x+b) ) is
A ( cdot frac{d^{2} y}{d x^{2}}+4 y=0 )
B. ( frac{d^{2} y}{d x^{2}}-4 y=0 )
c. ( frac{d^{2} y}{d x^{2}}+2 y=0 )
D. ( frac{d^{2} y}{d x^{2}}+y=0 )
12
211 A country has a food deficit of 10%. Its population grows
continously at a rate of 3% per year. Its annual food
production every year is 4% more than that of the last year.
Assuming that the average food requirement per person
remains constant, prove that the country will become self-
sufficient in food after n years, where n is the smallest integer
In 10-In 9
10002.(2000 – 10 Marks)
bigger than or equal to
12
212 Form the differential equation by eliminating arbitrary constants from the relation ( boldsymbol{A x}^{2}+boldsymbol{B y}^{2}=1 ) or ( frac{boldsymbol{x}^{2}}{boldsymbol{a}^{2}}+ )
( frac{boldsymbol{y}^{2}}{boldsymbol{b}^{2}}=mathbf{1} )
12
213 Form the differential equation of all family of lines ( boldsymbol{y}=boldsymbol{m} boldsymbol{x}+frac{boldsymbol{4}}{boldsymbol{m}} ) by
eliminating the arbitrary constant ‘ ( m^{prime} )
is
A ( cdot frac{d^{2} y}{d x^{2}}=0 )
в. ( quad xleft(frac{d y}{d x}right)^{2}-y frac{d y}{d x}+4=0 )
( ^{mathrm{c}} xleft(frac{d y}{d x}right)^{2}+y frac{d y}{d x}+4=0 )
D. ( frac{d y}{d x}=0 )
12
214 Find the particular solution of the difference equation
( left(1-y^{2}right)(1+log x) d x+2 x y d y=0 )
given that ( y=0 ) when ( x=1 )
12
215 If ( y=sqrt{frac{1-x}{1+x} text { then find }left(1-x^{2}right) frac{d y}{d x}+} )
( y= )
A . 1
B. –
( c cdot 2 )
D.
12
216 From the differential equation to the family of curves ( y=a e^{-2 x}+b e^{3 x} ) by
eliminating arbitrary constants ( a ) and ( b )
12
217 If the equation of family of curves be
( boldsymbol{y}=boldsymbol{a} cos (boldsymbol{x}+boldsymbol{b}), ) where ( a, boldsymbol{b} ) are arbitrary
constants,and ( boldsymbol{y}^{prime}=frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, boldsymbol{y}^{prime prime}=frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}^{prime}} ) etc
then their differential equation is :
A ( cdot y^{prime prime}=y )
B ( cdot y^{prime prime}=y y^{prime} )
C ( cdot y^{prime prime}+y=0 )
( mathbf{D} cdot y^{prime prime}=y+y^{prime} )
12
218 Verify that ( y^{2}=4 a(x+a) ) is a solution
of the differential equation
( boldsymbol{y}left{1-left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}right}=boldsymbol{2} boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
12
219 Degree of ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{3}+mathbf{3} boldsymbol{y}=boldsymbol{x}^{2} ) is
( mathbf{A} cdot mathbf{4} )
B. 2
( c .3 )
D.
12
220 The order and degree of the differential equation, ( left(frac{d^{2} y}{d x^{2}}right)^{3}=sin y+3 x quad ) are
A . 3,2
B . 2,3
( c .3, ) not defined
D. Not defined, 2
12
221 Let ( boldsymbol{y}=boldsymbol{a} cos (log boldsymbol{x})+boldsymbol{b} sin (log boldsymbol{x}) ) is a
solution of the differential equation ( boldsymbol{x}^{2} frac{d^{2} y}{d x^{2}}+boldsymbol{x} frac{d y}{d x}+boldsymbol{y}=0 . ) Prove
12
222 Solve the differential equation: ( left(x^{2}+right. )
( boldsymbol{x} boldsymbol{y}) boldsymbol{d} boldsymbol{y}=left(boldsymbol{x}^{2}+boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{x} )
A ( cdot k(y-x)^{2}=x e^{-y / x} / x )
B. ( k(y-x)^{2}=x e^{y / x} )
C. ( k(x-y)^{2}=x e^{-x / y} / y )
D. ( k(y-x)=-x e^{y / x} )
12
223 The differential equation for the family of circle
x2 + y2 – 2ay = 0, where a is an arbitrary constant is
[2004]
(a) (x2 + y²)y’ = 2xy (b) 2(x2 + y2)y’ = xy
(c) (x2 – y2)y’ = 2xy (d) 2(x2 – y2)y’ = xy
12.
12
224 Find the order and the degree of the
differential equation:
( left[1+left(frac{d y}{d x}right)^{2}right]^{frac{3}{2}}=5 frac{d^{2} y}{d x^{2}} )
12
225 If ( boldsymbol{y}=boldsymbol{A}+boldsymbol{B} boldsymbol{x}^{2} ) then:
A ( cdot frac{d^{2} y}{d x^{2}}=2 x y )
B. ( x frac{d^{2} y}{d x^{2}}=y_{1} )
c. ( x frac{d^{2} y}{d x^{2}}-frac{d y}{d x}+y=0 )
D. ( x frac{d^{2} y}{d x^{2}}+frac{d y}{d x}+y=0 )
12
226 ( y^{2}=6 x^{3}+x-8 ) find ( frac{d y}{d x} ) 12
227 Solve:
( (x+y) frac{d y}{d x}=1 )
12
228 The number of arbitarary constants in the solution of a differential equation of
degree 2 and order 3 is:
( A cdot 2 )
B. 3
( c cdot 5 )
D.
12
229 Find the order and degree of ( [1+ ) ( left.boldsymbol{y}^{prime}^{2}right]^{1 / 2}=boldsymbol{x}^{2}+boldsymbol{y} )
A . 1,2
в. 2,1
c. 1,1
D. 2,2
12
230 The degree and order of the differential equation of the family of all those
parabola’s whose axis is ( x ) -axis are
respectively.
A ( cdot 1,2 )
B. 3,2
c. 2,3
D. 2,
12
231 Let ( boldsymbol{y}=boldsymbol{y}(boldsymbol{x}) ) be the solution of the
differential equation, ( left(x^{2}+1right)^{2} frac{d y}{d x}+ )
( 2 xleft(x^{2}+1right) y=1 ) such that ( y(0)=0 )
( sqrt{boldsymbol{a} boldsymbol{y}}(1)=frac{boldsymbol{pi}}{mathbf{3 2}}, ) then the value of ‘a’ is:
A ( cdot frac{1}{2} )
B. ( frac{1}{16} )
( c cdot frac{1}{4} )
D.
12
232 A body is heated at ( 110^{circ} mathrm{C} ) and placed in
air at ( 10^{circ} mathrm{C} ). After 1 hour its
temperature is ( 60^{circ} mathrm{C} ). How much
additional time is required for it to cool
to ( 35^{circ} ) C?
12
233 Which one of the following differential equations represents the family of straight lines which are at unit distance from the origin?
( ^{A} cdotleft(y-x frac{d y}{d x}right)^{2}=1-left(frac{d y}{d x}right)^{2} )
( ^{mathrm{B}}left(y+x frac{d y}{d x}right)^{2}=1+left(frac{d y}{d x}right)^{2} )
( ^{mathrm{c}}left(y-x frac{d y}{d x}right)^{2}=1+left(frac{d y}{d x}right)^{2} )
( ^{mathrm{D} cdot}left(y+x frac{d y}{d x}right)^{2}=1-left(frac{d y}{d x}right)^{2} )
12
234 The equation of the curve through ( left(0, frac{pi}{4}right) ) satisfying the differential
equation. ( e^{x} tan y d x+(1+ )
( left.e^{x}right) sec ^{2} y d y=0 ) is given by
A ( cdotleft(1+e^{x}right) tan y=2 )
B . ( 1+e^{x}=2 tan y )
C ( cdot 1+e^{x}=2 sec y )
D・ ( left(1+e^{x}right) tan y=1 )
12
235 ( (x+y)(d x-d y)=d x+d y )
Solve the above equation.
12
236 The solution of the equation ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}= )
( e^{x}+e^{-x} ) is –
Note : (where c ( & ) d are arbitrary constants in the given options)
A ( cdot y=e^{x}-e^{-x}+c x+d )
B . ( y=e^{x}+e^{-x}+c x+d )
c. ( y=-e^{x}+e^{-x}+c x+d )
D. None of these
12
237 A fossilised bone is found to contain
( 0.1 % ) of its original C14. Find the age of the fossil.
A. 57100 years
B. 43100 years
c. 27860 years
D. None of these
12
238 The solution of the D.E. ( left(x^{3}-right. )
( left.mathbf{3} boldsymbol{x} boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{x}=left(boldsymbol{y}^{3}-boldsymbol{3} boldsymbol{x}^{2} boldsymbol{y}right) boldsymbol{d} boldsymbol{y}, ) is:
A ( cdot y^{2}-x^{2}=cleft(y^{2}+x^{2}right)^{2} )
B . ( y^{2}-x^{2}=left(y^{2}+x^{2}right)^{2} )
C ( cdot y^{2}+x^{2}=cleft(y^{2}-x^{2}right)^{2} )
D. ( cleft(y^{2}+x^{2}right)=left(y^{2}-x^{2}right)^{2} )
12
239 The D. E of the family of all circles in the first quadrant touching the coordinate
axes
( mathbf{A} cdotleft[1+y_{1}^{2}right]^{3}=r^{2} y_{2}^{2} )
B ( cdotleft[1+y_{2}right]^{3}=r^{2} y_{1}^{2} )
C ( cdotleft[1+y_{1}^{2}right]^{2}=r^{2} y_{2}^{3} )
D. ( left[1+y_{1}right]^{3}=r^{2} y_{2}^{2} )
12
240 The number of arbitrary constants in the particular solution of the differential equation of order 3 is
( A cdot 0 )
B.
( c cdot 2 )
( D )
12
241 Solution of the differential equation ydx + (x + xy)dy = 0
is
[2004]
(a)
log y = Cx
— + log y = C
xy
= C
(c)
—+log y=C
xy
Xy
12
242 Assertion
The differential equation of the family of
curves represented by ( boldsymbol{y}=left(boldsymbol{a}+boldsymbol{b} e^{c}right) boldsymbol{x} )
is ( x frac{d y}{d x}-y=0, ) where ( a, b ) and ( c ) are
constant.
Reason
In general, a differential equation of ( n ) th
order is obtained on eliminating ( n ) arbitrary constants.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
243 A resistance of ( 100 Omega ) and inductance ( L )
henry are connected in series with a
battery of 20 volts.The current at any instant, if the relation between is ( boldsymbol{L}, boldsymbol{R}, boldsymbol{E} ) is ( boldsymbol{L} frac{boldsymbol{d} boldsymbol{i}}{boldsymbol{d} boldsymbol{t}}+boldsymbol{R} boldsymbol{i}=boldsymbol{E} ) is given by?
A ( . i=0.6left(-e^{-200 t}right) )
В ( cdot i=0.4left(-e^{-200 t}right) )
c. ( quad i=0.2left(1-e^{frac{-100}{L} t}right) )
D . ( i=0.8left(-e^{-200 t}right) )
12
244 If ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x} boldsymbol{y}}{boldsymbol{x}^{2}+boldsymbol{y}^{2}} ; boldsymbol{y}(1)=1 ; ) then a value
of ( boldsymbol{x} ) satisfying ( boldsymbol{y}(boldsymbol{x})=boldsymbol{e} ) is
A ( cdot sqrt{3} e )
B. ( frac{e}{sqrt{2}} )
( c cdot sqrt{2} e )
D. ( frac{1}{2} sqrt{3} epsilon )
12
245 Consider the differential equation:
( frac{left(1+left(frac{d y}{d x}right)^{2}right)^{frac{3}{2}}}{frac{d^{2} y}{d x^{2}}}=c )
Find order and degree.
12
246 Form the differential equation
corresponding to ( y=e^{m x} ) by
eliminating ( boldsymbol{m} )
12
247 Solution of the differential equation
( x d y-y d x=0 ) represents.
A. A parabola whose vertex is at origin.
B. A circle whose centre is at origin
c. A rectangular hyperbola.
D. A straight line passing thorugh origin.
12
248 The solution of differential equation
( x cos ^{2} y d x=y cos ^{2} x d y ) is
A ( cdot x tan x-y tan y-log left(frac{sec x}{sec y}right)=c )
B. ( y tan x-x tan y-log (sec x cdot sec y)=c )
c. ( x tan x-y tan y+log (sec x cdot sec y)=c )
D. None of the above
12
249 For the following differential equation, find the general solution. ( frac{d y}{d x}+x=1 ) 12
250 The differential equation representing the family of curves ( y^{2}=2 c(x+sqrt{c}) )
where ( c ) is a positive parameter is of
A . order 3
B. order 2
c. degree 3
D. degree 4
12
251 The equation of the curve whose slope at any point is equal to ( y+2 x ) and
which passes through the origin is
12
252 ( (1+x y) frac{d y}{d x}+y^{3}=0 y(0)=1 ) Find ( k ) if
the constant in the solution is ( frac{k}{e} )
12
253 Solve:
( frac{d y}{d x}=x+1 ; ) find ( y ) when ( x=2 )
12
254 Find the differential equation of all the
ellipse whose center at origin and axis are along the coordinate axis.
12
255 Solution of the differential equation ( boldsymbol{y}^{prime}+boldsymbol{y} sec ^{2} boldsymbol{x}=sec ^{2} boldsymbol{x} cdot boldsymbol{operatorname { t a n }} boldsymbol{x} ) is
A ( cdot y=(tan x+1)+c e^{-tan x} )
B . ( y=(tan x-1)+c e^{-t a n x} )
C ( cdot y=(tan x+1)+c e^{tan x} )
D. None of these
12
256 Show that ( x y=a e^{x}+b e^{-x}+x^{2} ) is a
solution of the differential equation ( x frac{d^{2} y}{d x^{2}}+2 frac{d y}{d x}-x y+x^{2}-2=0 )
12
257 The differential equation of all circles passing through the origin and having their centres on the ( x ) -axis is
A ( cdot y^{2}=x^{2}+2 x y frac{d y}{d x} )
B. ( y^{2}=x^{2}-2 x y frac{d y}{d x} )
c. ( x^{2}=y^{2}+x y frac{d y}{d x} )
D. ( x^{2}=y^{2}+3 x y frac{d y}{d x} )
12
258 Write the degree of the differential
equation ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+boldsymbol{x}left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}= )
( 2 x^{2} log left(frac{d^{2} y}{d x^{2}}right) )
12
259 Let I be the purchase value of an equipment and ( V(t) ) be the value after it has been used for t years. The value ( V(t) ) depreciates at a rate given by differential equation ( frac{boldsymbol{d} boldsymbol{V}(boldsymbol{t})}{boldsymbol{d} boldsymbol{t}}=-boldsymbol{k}(boldsymbol{T}- )
( t), ) where ( k>0 ) is a constant and T is
the total life in years of the equipment. Then the scrap value ( V(t) ) of the
equipment is?
A ( cdot_{I-frac{k(T-t)^{2}}{2}} )
( frac{k(T-t)^{2}}{2}+I )
( c cdot T^{2}-frac{I}{k} )
D. ( _{I-frac{k T^{2}}{2}} )
12
260 Solution of ( (x+y)^{2} frac{d y}{d x}=a^{2}left(^{prime} a^{prime} ) being a right.
constant) is:
A ( cdot frac{(x+y)}{a}=tan frac{y+c}{a}, c ) is an arbitrary constant
B. ( x y=a tan c x, c ) is an arbitrary constant
c. ( frac{x}{a}=tan frac{y}{c}, c ) is an arbitrary constant
D. ( x y=tan (x+c), c ) is an arbitrary constant
12
261 ( y=tan ^{-1} frac{sin x}{1+cos x}, ) then find ( 4 frac{d y}{d x} ) 12
262 The solution of ( frac{d y}{d x}=frac{x+x^{2}}{y+y^{2}} ) is:
A ( cdot x^{3}-y^{3}-y^{2}-x^{2}=c )
B . ( 2left(x^{3}-y^{3}right)+3left(x^{2}-y^{2}right)=c )
c. ( x^{2}+y^{2}+x+y=c )
D. ( x^{2} y+x y^{2}=c )
12
263 The order and degree of
( left[frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}right]^{1 / 2}=frac{d^{3} y}{d x^{3}} ) is:
( A cdot 1,2 )
B. 3,
c. 3,2
D. 2,
12
264 Find the particular solution of the differential equation ( (x-y) frac{d y}{d x}=x+ )
( 2 y, ) given that when ( x=1, y=0 )
12
265 Solve the differential equation:
( sqrt{1+x^{2}} d x+sqrt{1+y^{2}} d y=0 )
A ( . x sqrt{1+x^{2}}+y sqrt{1+y^{2}}+ )
( log [(x+sqrt{1+x^{2}})(y+sqrt{1+y^{2}})]=c )
B . ( sqrt{1+x^{2}}+sqrt{1+y^{2}}=c )
c. ( frac{1}{sqrt{1+x^{2}}}+frac{1}{sqrt{1+y^{2}}}=c )
D ( cdot log {(sqrt{1+x^{2}})+(sqrt{1+y^{2}})}=x+c )
12
266 Find the differential equation of the
family of concentric circles ( x^{2}+y^{2}= )
( boldsymbol{a}^{2} )
12
267 Form the differential equation of the family of circles in the first quadrant, which touches the coordinate axes. 12
268 The solution of the differential equation ( boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{2} boldsymbol{y}=boldsymbol{x}^{2}(boldsymbol{x} neq mathbf{0}) ) with ( boldsymbol{y}(mathbf{1})=mathbf{1} )
is?
A ( cdot y=frac{x^{3}}{5}+frac{1}{5 x^{2}} )
B. ( y=frac{4}{5} x^{3}+frac{1}{5 x^{2}} )
c. ( y=frac{3}{4} x^{2}+frac{1}{4 x^{2}} )
D. ( y=frac{x^{2}}{4}+frac{3}{4 x^{2}} )
12
269 Determine the equation of the curve passing through the origin in the form ( y ) ( =f(x), ) which satisfies the differential equation ( frac{d y}{d x}=sin (10 x+6 y) ) 12
270 Solve:
(i) ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{e}^{boldsymbol{3} boldsymbol{x}-boldsymbol{2} boldsymbol{y}}+boldsymbol{x}^{2} boldsymbol{e}^{-boldsymbol{2} boldsymbol{y}} )
12
271 Find the Particular solution of the
differential equations
( x^{2} d y+left(x y+y^{2}right) d x=0 ; y=1 ) when
( boldsymbol{x}=mathbf{1} )
12
272 The equation of motion of a body falling under gravity is given by ( frac{d v}{d t}=g- ) ( frac{boldsymbol{g}}{boldsymbol{lambda}^{2}} boldsymbol{v}^{2} . ) Distance travelled as a function
of time is given by?. Give at ( boldsymbol{v}=mathbf{0}, boldsymbol{t}=mathbf{0} )
A ( cdot x=frac{lambda^{2}}{g} log sinh left(frac{g t}{lambda}right) )
B. ( x=frac{lambda}{g} log cos hleft(frac{g t}{lambda^{2}}right) )
( ^{mathbf{C}} x=frac{lambda^{2}}{g} log sec hleft(frac{g t}{lambda}right) )
D ( x=frac{lambda^{2}}{g} log cos hleft(frac{g t}{lambda}right) )
12
273 The differential equation which
represents the family of curves given by
( tan y=cleft(1-e^{x}right) ) is
A ( cdot e^{x} tan y d x+left(1-e^{x}right) d y=0 )
B . ( e^{x} ) tan ( y d x+left(1-e^{x}right) sec ^{2} y d y=0 )
C ( cdot e^{x}left(1-e^{x}right) d x+tan y d y=0 )
D. ( e^{x} tan y d y+left(1-e^{x}right) d x=0 )
12
274 Order and degree of a differential
equation ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}=left{boldsymbol{y}+left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}right}^{1 / 4} ) are
A. 4 and 2
B. 1 and 2
( c cdot 1 ) and 4
D. 2 and 4
12
275 Solution of differential equation ( frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{d} boldsymbol{y}}= )
( tan x(1+y sin x) ) is given by
A ( cdot operatorname{cosec} x=-y+1+C e^{-y} )
B. ( y=tan x+C e^{X} )
c. ( sin x e^{y}=1+y+C )
D. ( operatorname{cosec} x=y+C e^{y} )
12
276 Find the differential equation of the family of all the circles.
(A) touching X-axis at the origin.
(B) touching Y-axis at the origin.
12
277 A particle starts at the origin and moves along the ( x ) -axis in such a way
that its velocity at the point ( (x, 0) ) is
given by the formula ( frac{d x}{d t}=cos ^{2} pi x )
Then the particle never reaches the point on:
A ( cdot x=frac{1}{4} )
в. ( _{x}=frac{3}{4} )
c. ( _{x}=frac{1}{2} )
D. ( x=1 )
12
278 Assertion
A normal is drawn at a point ( boldsymbol{P}(boldsymbol{x}, boldsymbol{y}) ) of ( mathbf{a} ) curve. It meets the ( x ) -axis and the ( y ) -axis in point ( A ) and ( B ), respectively, such that ( frac{1}{O A}+frac{1}{O B}=1, ) where ( O ) is the
origin. The equation of such a curve passing through ( (mathbf{5}, mathbf{4}) ) is ( (x-1)^{2}+ )
( (y-1)^{2}=25 )
Reason ( boldsymbol{O A}=boldsymbol{x}+boldsymbol{y} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) and ( boldsymbol{O} boldsymbol{B}=frac{boldsymbol{x}+boldsymbol{y} frac{d boldsymbol{y}}{d boldsymbol{x}}}{frac{d boldsymbol{y}}{d boldsymbol{x}}} )
A. Both Assertion and Reason are correct and Reason the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
279 6.
Let y(x) be a solution of the differential equation
(1+e y’+ yet =1. If y(0)=2, then which of the following
statement is (are) true?
(JEE Adv. 2015)
(a) y(-4)=0
(b) y(-2)=0
(C) y(x) has a critical point in the interval (-1,0)
(d) y(x) has no critical point in the interval (-1,0)
12
280 ( mathbf{f} boldsymbol{y}=left(sin ^{-1} xright)^{2}, ) then prove that
( left(1-x^{2}right) frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}-2=0 )
12
281 Form a differential equation representing the given family of curves by eliminating arbitrary constant a and b.
( y=e^{x}(a cos x+b sin x) )
12
282 The degree of the differential equation ( boldsymbol{x}=mathbf{1}+left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)+frac{mathbf{1}}{mathbf{2 !}}left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}+ )
( frac{1}{3 !}left(frac{d y}{d x}right)^{3}+dots )
A . 3
B . 2
( c cdot 1 )
D. not defined
12
283 The differential equation of the family of hyperbolas having centres at the origin and whose axes are the co-ordinate
axes is
A ( cdot x y y_{2}+x y_{1}^{2}+y y_{1}=0 )
B. ( x y y_{2}-x y_{1}^{2}-y y_{1}=0 )
c. ( x y y_{2}+x y_{1}^{2}-y y_{1}=0 )
D. ( x y y_{2}+x y_{1}-y y_{1}=0 )
12
284 Form the differential equation by
eliminating the arbitrary constant ( a ) from the relation ( (x-a)^{2}+y^{2}=1 )
12
285 The differential equation of system of concentric circles with centres (1,2) is
A ( cdot frac{d y}{d x}=frac{x-1}{x-2} )
B. ( frac{d y}{d x}=frac{x-1}{2-x} )
c. ( frac{d y}{d x}=frac{x-1}{2-y} )
D. ( frac{d y}{d x}=frac{x-1}{y-2} )
12
286 Equation of the curve whose polar sub ( operatorname{tangent} r^{2} frac{d theta}{d r} ) is constant
A. ( r(theta+c)+k=0 )
B ( cdot r^{2}(theta+c)=2 k )
c. ( r(theta-c)=k^{2} )
D. ( r theta=c )
12
287 Solve for differntial equation:
( left(x^{3}-xright) frac{d y}{d x}-left(3 x^{2}-1right) y=x^{5}- )
( 2 x^{3}+x )
A ( cdot y frac{1}{xleft(x^{2}+1right)}=log x+c )
в. ( y cdot frac{1}{xleft(x^{2}-1right)}=-log x+c )
c. ( y frac{1}{xleft(x^{2}-1right)}=log x+c )
D. None of these.
12
288 Solve the differential equation:
( left(x y^{2}-e^{1 / x^{3}}right) d x-x^{2} y d y=0 )
A ( cdot 3 y^{2}=2 x^{2} e^{1 / x^{3}}+c x^{2} )
B . ( 3 y^{2}=2 x^{2} e^{1 / x^{3}}-c x^{2} )
c. ( 3 y^{2}=-2 x^{2} e^{1 / x^{3}}+c x^{2} )
D. None of these.
12
289 Solve:
( boldsymbol{x}^{e}+boldsymbol{e}^{boldsymbol{x}}+boldsymbol{e}^{boldsymbol{e}} )
12
290 A right circular cone with radius R and height H contains a
liquid which evaporates at a rate proportional to its surface
area in contact with air (proportionality constant = k > 0).
Find the time after which the cone is empty.
12
291 The order of the differential equation
whose general solution is ( y= )
( c, cos 2 x+c_{2} cos ^{2} x+c_{3} sin ^{2} x+c_{4} )
( A cdot 2 )
B. 4
( c .3 )
D. None of these
12
292 Solve:
( left(3 x^{2}+y^{2}right) d y+left(x^{2}+3 y^{2}right) d x=0 )
12
293 ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}+boldsymbol{y}+mathbf{1}}{mathbf{2} boldsymbol{x}+mathbf{2} boldsymbol{y}+mathbf{3}}, ) its solution is ( boldsymbol{x}+ )
( y+frac{k}{3}=c e^{3(x-2 y)}, ) what is ( k )
( A )
B. 2
( c cdot 5 )
D. 4
12
294 ff ( p^{2}=a^{2} cos ^{2} theta+b^{2} sin ^{2} theta ) then Prove
That:
( boldsymbol{P}+frac{boldsymbol{d}^{2} boldsymbol{p}}{boldsymbol{d} boldsymbol{theta}^{2}}=frac{boldsymbol{a}^{2} boldsymbol{b}^{2}}{boldsymbol{p}^{3}} )
12
295 The differential equation by eliminating the arbitary constants from the equation ( boldsymbol{y}=boldsymbol{a} cos boldsymbol{x}+boldsymbol{b} sin boldsymbol{x}+boldsymbol{x} sin boldsymbol{x} )
is
( mathbf{A} cdot y_{2}+y=3 cos x )
В. ( y_{2}+2 y=2 cos x )
C ( cdot y_{2}+y=2 cos x )
D. ( y_{2}+y=4 cos x )
12
296 The differential equation representing
the family of curves ( y^{2}=a(a x+b) )
where ( a ) and ( b ) are arbitrary constants, is of
A. order 1, degree 1
B. order 1, degree 3
c. order 2 ,degree 3
D. order ( 1, ) degree 4
E. order 2, degree 1
12
297 Solve the differential equation : ( boldsymbol{y}+ )
( boldsymbol{x} frac{d boldsymbol{y}}{d boldsymbol{x}}=boldsymbol{x}-boldsymbol{y} frac{d boldsymbol{y}}{d boldsymbol{x} boldsymbol{x}} )
12
298 Order and degree of ( left(frac{d y}{d x}right)^{2}-5 y=3 cos x )
are:
A . 2,3
B. 1,2
( c cdot 2,2 )
D. 1,1
12
299 The solution of the D.E ( y y_{1}= )
( left[frac{boldsymbol{y}^{2}}{boldsymbol{x}^{2}}+frac{boldsymbol{f}left(boldsymbol{y}^{2} / boldsymbol{x}^{2}right)}{boldsymbol{f}^{prime}left(boldsymbol{y}^{2} / boldsymbol{x}^{2}right)}right] ) is:
( ^{A} cdot fleft(frac{y^{2}}{x^{2}}right)=c x^{2} )
B. ( x^{2} fleft(frac{y^{2}}{x^{2}}right)=c^{2} y^{2} )
C ( cdot x^{2} fleft(y^{2} / x^{2}right)=c )
D. ( fleft(frac{y^{2}}{x^{2}}right)=c y / x )
12
300 Let u(x) and v(x) satisfy the differential equation”
+p(x) u
dx
= f(x) and av +p(x) v = g(x), where p(x) f(x) and g(x) are
dx
continuous functions. If u(x) > v(x) for some x, and
· f(x) > g(x) for all x>xy, prove that any point (x, y) where x>
X,, does not satisfy the equatons y= u(x) and y = v(x).
(1997 – 5 Marks)
12
301 The differential equation of all circles passing through the origin and having their centres on the X-axis, is
A ( x^{2}=x^{2}+2 x y frac{d y}{d x} )
B. ( _{y^{2}}=x^{2}-2 x y frac{d y}{d x} )
c. ( _{y^{2}=x^{2}+x y frac{d y}{d x}} )
D. ( y^{2}=x^{2}+3 x y frac{d y}{d x} )
12
302 Find the age of an object that has been excavated and found to have ( 90 % ) of its
original amount of radioactive Carbon
14
A. 378 years old
B. 248 years old
c. 878 years old
D. None of these
12
303 Solve ( left(2 x-10 y^{3}right) frac{d y}{d x}+y=0 )
A ( cdot x=-2 y^{3}+c y^{-2} )
B . ( x=2 y^{3}+c y^{-2} )
c. ( x=2 y^{3}+c y^{2} )
D. None of these
12
304 ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+frac{boldsymbol{y}}{boldsymbol{x}}=frac{boldsymbol{y}^{2}}{boldsymbol{x}^{2}} )
Solve
( mathbf{A} cdot y-2 x=k x^{2} y )
B ( cdot y+2 x=k x^{2} y )
C ( cdot 2 y-x=k x^{2} y )
D. ( 2 y+x=k x^{2} y )
12
305 Through any point, ( (x, y) ) of a curve passing through the origin, lines are drawn parallel to the coordinate axes.
The curve divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other.
Then the curve may represent a family of:
A . circles
B. parabolas
c. ellipses
D. hyperbolas
12
306 Find a differential equation
corresponding to ( y=a x^{2}+b x )
12
307 The solution of the differential equation,
( boldsymbol{e}^{boldsymbol{x}}(boldsymbol{x}+mathbf{1}) boldsymbol{d} boldsymbol{x}+left(boldsymbol{y} boldsymbol{e}^{boldsymbol{y}}+boldsymbol{x} boldsymbol{e}^{boldsymbol{x}}right) boldsymbol{d} boldsymbol{y}=mathbf{0} )
with initial condition ( f(0)=0, ) is-
A ( cdot x e^{x}+2 y^{2} e^{y}=0 )
В ( cdot 2 x e^{x}+y^{2} e^{y}=0 )
C. ( x e^{x}-2 y^{2} e^{y}=0 )
D. ( 2 x e^{x}-y^{2} e^{y}=0 )
12
308 dy
It &
= y (log y – log x + 1), then the solution of the
dx
equation is
[2005]
(a), y log() = ex
(C) log (9) =a
(b) xlog (3) =cy
(a) log (*) –cy
y
12
309 Solution of differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}- )
( 2 x y=x ) is
A ( cdot y=C e^{x^{2}}-frac{1}{2} )
B. ( y=C e^{x^{2}}+frac{1}{2} )
c. ( y=C x^{2}-frac{1}{2} )
D. None
12
310 The number of arbitrary constant in the general solution of differential equation of order 3 is
( mathbf{A} cdot mathbf{0} )
B . 2
( c .3 )
D.
12
311 Family of curves ( boldsymbol{y}=e^{x}(boldsymbol{A} cos boldsymbol{x}+ )
( B sin x), ) represents the differential
equation?
A ( cdot frac{d^{2} y}{d x^{2}}=2 frac{d y}{d x}-y )
B. ( frac{d^{2} y}{d x^{2}}=2 frac{d y}{d x}-2 y )
( ^{mathbf{C}} cdot frac{d^{2} y}{d x^{2}}=frac{d y}{d x}-2 y )
D. None of the above
12
312 The solution to the differential equation ( (x+1) frac{d y}{d x}-y=e^{3 x}(x+1)^{2} ) is
A ( cdot y=(x+1) e^{3 x}+c )
B. ( 3 y=(x+1)+e^{3 x}+c )
c. ( frac{3 y}{x+1}=e^{3 x}+c )
D. ( y e^{-3 x}=3(x+1)+c )
12
313 The solution of ( frac{d y}{d x}=frac{sqrt{x^{2}-y^{2}}+y}{x} ) is:
( mathbf{A} cdot tan ^{-1}left(frac{y}{x}right)=log (c x) )
B・sin ( ^{-1}left(frac{y}{x}right)=log (c x) )
c. ( cos ^{-1}left(frac{y}{x}right)=log (c y) )
D ( cdot sec ^{-1}left(frac{y}{x}right)=log (c y) )
12
314 Find the differential equation of family of circles of all of radius ( 5, ) with their
centres on the y-axis.
12
315 Solve:
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{(boldsymbol{x}-boldsymbol{y})+boldsymbol{3}}{boldsymbol{2}(boldsymbol{x}-boldsymbol{y})+boldsymbol{5}} )
12
316 Solution of differential equation ( frac{d y}{d x}+ ) ( frac{boldsymbol{y}}{mathbf{1}+boldsymbol{x}^{2}}=frac{e^{t a n^{-1} boldsymbol{x}}}{1+boldsymbol{x}^{2}} ) is
A ( quad y=frac{e^{tan ^{-1} x}}{2}+c )
B. ( y=frac{e^{2 t a n^{-1} x}}{2}+c )
c. ( y=frac{e^{tan ^{-1} x}}{2}+c e^{-tan ^{-1} x} )
D. ( y=e^{tan ^{-1} x}+c )
12
317 Find the differential equation representing the family of curves ( boldsymbol{y}= )
( a e^{b x+5}, ) where a and ( b ) are arbitrary
constants.
12
318 What is the solution of the differential
equation ( frac{y d x-x d y}{y^{2}}=0 ? )
where ( c ) is an arbitrary constant.
A ( . x y=c )
в. ( y=c x )
c. ( x+y=c )
D. ( x-y=c )
12
319 ( left(x^{2}+y^{2}right) d x+2 x y d y=0 )
Solve the above equation.
A ( cdot xleft(x^{2}+3 y^{2}right)=c )
B. ( xleft(3 x^{2}+y^{2}right)=c )
C ( cdot yleft(3 x^{2}+y^{2}right)=c )
D. ( yleft(x^{2}+3 y^{2}right)=c )
12
320 Solve the following differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}^{2}+boldsymbol{3} boldsymbol{x}+boldsymbol{7} ) 12
321 Write the degree of the differentia
quation ( :left(frac{d y}{d x}right)^{4}+3 yleft(frac{d^{2} y}{d x^{2}}right)=0 )
12
322 Prove that ( x^{2}-y^{2}=cleft(x^{2}+y^{2}right)^{2} ) is the
general solution of differential equation
( left(x^{3}-3 x y^{2}right) d x=left(y^{3}-3 x^{2} yright) d y )
where ( c ) is a parameter
12
323 The equation of electromotive force in
terms of current ( i ) for an electrical
circuit having resistance ( boldsymbol{R} ) and condenser ( C ) in series is ( boldsymbol{E}=boldsymbol{R} boldsymbol{i}+ )
( int frac{i}{c} d t . ) The current at any time ( t, ) when
( boldsymbol{E}=boldsymbol{E}_{0} sin omega boldsymbol{t} ) is?
A ( cdot i=frac{E_{0} omega c}{sqrt{1+R^{2} c^{2} omega^{2}}} sin (omega t-phi)+k e^{-t / R c} )
B. ( i=frac{E_{0} omega c}{sqrt{1+R^{2} c^{2} omega^{2}}} cos (omega t-phi)+k e^{-t / R c} )
c. ( _{i}=frac{E_{0} omega c}{sqrt{1+R^{2} c^{2} omega^{2}}} cosh (omega t-phi)+k e^{-t / R c} )
D. ( i=frac{E_{0} omega c}{sqrt{1+R^{2} c^{2} omega^{2}}} sinh (omega t-phi)+k e^{-t / R c} )
12
324 Solve the differential equation:
( cos x frac{d y}{d x}+y=sin x )
12
325 Form the differential equation representing the family of curves ( y= )
( a e^{2 x}+b e^{-2 x} ) where a and ( b ) are arbitrary
constants.
12
326 Let ( y=x e^{-x} ) then prove that ( x frac{d y}{d x}= ) ( (1-x) y ) 12
327 [
begin{aligned}
text { If: } boldsymbol{f}(boldsymbol{x})=& frac{boldsymbol{x}left(e^{1 / x}-e^{-1 / x}right)}{e^{1 / x}+e^{-1 / x}}, ldots x neq 0 \
&=mathbf{0}, quad ldots . . x=0
end{aligned}
]
then ( boldsymbol{f}(boldsymbol{x}) ) is
A. continuous everywhere but not differentiable at ( x=0 )
B. continuous and differentiable everywhere
c. discontinuous at ( x=0 )
D. none of these
12
328 ( boldsymbol{F}(boldsymbol{x}, boldsymbol{y})=boldsymbol{x}^{3}+boldsymbol{y}^{3}+boldsymbol{3} boldsymbol{x}^{2} boldsymbol{y}+boldsymbol{3} boldsymbol{x} boldsymbol{y}^{2} cdot mathrm{I} )
this homogeneous function
12
329 The order and degree of the differential equation of all circles in the first quadratic which touch the co-ordinate axis is:
A .1,2
в. 2,1
( c .3,2 )
D. 4,3
12
330 The D.E whose solution is ( y=left(c_{1} x+right. )
( left.c_{2}right) e^{5 x}: )
A ( cdot y_{2}+10 y_{1}+24 y=0 )
B ( cdot y_{2}-10 y_{1}+25 y=0 )
c. ( y_{2}-5 y_{1}+25 y=0 )
D. ( y_{2}-5 y_{1}+10 y=0 )
12
331 Solve the following differential ( left(e^{x}+1right) d y=(y+1) e^{x} d x ) 12
332 The D.E of the family of rectangular hyperbolas which have the coordinate
axes as asymptotes is:
A. ( x y_{1}+y=0 )
в. ( x y_{2}+y_{1}=0 )
( mathbf{c} cdot x y_{2}=y )
D. ( x y_{2}+y=0 )
12
333 Write the differential equation
representing family of curves ( boldsymbol{y}=boldsymbol{m} boldsymbol{x} )
where ( m ) is arbitrary constant.
12
334 The solution to the differential equation
( boldsymbol{y} ln boldsymbol{y}+boldsymbol{x} boldsymbol{y}^{prime}=mathbf{0} ) where ( boldsymbol{y}(mathbf{1})=boldsymbol{e}, ) is:
A. ( x(ln y)=1 )
в. ( x y(ln y)=1 )
c. ( (ln y)^{2}=2 )
D. ( ln y+left(frac{x^{2}}{2}right) y=1 )
12
335 Form the differential equation
corresponding to ( y=e^{m x} ) by
eliminating m.
12
336 Find the order and degree of the given
differential equation: ( y^{prime}=sin y . ) The
order of this equation is the same as its degree. If true enter 1 else enter 0
12
337 19.
Let y(x) be the solution of the differential equation
(x log x)
dx
+ y = 2x log x,(x 21). Then y(e) is equal to:
[JEE M 2015]
(6) 2e (c) e
(d) 0
(a) 2
12
338 The solution of the given D.E ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( frac{boldsymbol{y}+sqrt{boldsymbol{x}^{2}+boldsymbol{y}^{2}}}{boldsymbol{x}} ) is
A ( cdot y+sqrt{x^{2}+y^{2}}=c x^{2} )
B . ( x+sqrt{x^{2}+y^{2}}=c y^{2} )
c. ( y+sqrt{x^{2}+y^{2}}=c x )
D. ( x+sqrt{x^{2}+y^{2}}=c y )
12
339 A function ( y=f(x) ) satisfying the
differential equation ( frac{d y}{d x} cdot sin x- )
( y cos x+frac{sin ^{2} x}{x^{2}}=0 ) such that ( y rightarrow 0 ) as
( x rightarrow infty ) then the correct statement
which is correct is
( mathbf{A} cdot lim _{x rightarrow 0} i t f(x)=1 )
B. ( int_{0}^{pi / 2} f(x) d x ) is less than ( frac{pi}{2} )
c. ( int_{0}^{pi / 2} f(x) d x ) is greater than unit
D cdot ( f(x) ) is an odd function
12
340 The differential equation of the family of
curves ( r^{2}=a^{2} cos 2 theta ) where ‘a’ is
arbitrary constant is:
This question has multiple correct options
A ( cdot frac{d r}{d theta}=-r tan 2 theta )
B. ( frac{d r}{d theta}=r cot 2 theta )
c. ( frac{d r}{d theta} cos 2 theta+r sin 2 theta=0 )
D. ( frac{d r}{d theta}=0 )
12
341 ff ( boldsymbol{y}=frac{a}{2}left(e^{frac{x}{a}}+e^{frac{-x}{a}}right) ) and ( frac{d^{2} y}{d x^{2}}=y, ) then a
equals
A .
в.
( c cdot frac{1}{3} )
D. ( frac{1}{sqrt{2}} )
12
342 The order of the differential equation whose general solution is given by ( boldsymbol{y}=left(boldsymbol{C}_{1}+boldsymbol{C}_{2}right) cos left(boldsymbol{x}+boldsymbol{C}_{3}right)-boldsymbol{C}_{4} boldsymbol{e}^{boldsymbol{x}+boldsymbol{C}_{5}} )
where, ( C_{1}, C_{2}, C_{3}, C_{4}, C_{5} ) are arbitrary constants, is
( mathbf{A} cdot mathbf{5} )
B. 4
( c .3 )
D.
12
343 If ( y=A e^{m x}+B e^{n x}, ) show that ( frac{d^{2} y}{d x^{2}}- )
( (boldsymbol{m}+boldsymbol{n}) frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{m} boldsymbol{n} boldsymbol{y}=mathbf{0} )
12
344 Solve ( (boldsymbol{x}+boldsymbol{y})^{2} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{a}^{2} ) 12
345 A hemispherical tank of radius 2 metres is initially full of
water and has an outlet of 12 cm2 cross-sectional area at
the bottom. The outlet is opened at some instant. The flow
through the outlet is according to the law v(t)=0.
6 2gh(t),
where v(t) and h(t) are respectively the velocity of the flow
through the outlet and the height of water level above the
outlet at time t, and g is the acceleration due to gravity. Find
the time it takes to empty the tank. (Hint: Form a differential
equation by relating the decrease of water level to the
outflow).
(2001 – 10 Marks)
12
346 D.E whose solution is ( boldsymbol{y}= )
( e^{x}left(c_{1} cos 2 x+c_{2} sin 2 xright): )
( mathbf{A} cdot y_{2}-2 y_{1}+5 y=0 )
B . ( y_{2}-6 y_{1}+4 y=0 )
c. ( y_{2}-5 y_{1}+2 y=0 )
D. ( y_{2}-y_{1}+4 y=0 )
12
347 1.
A solution of the differential equation
(1999 – 2 Marks)
dy
+
y = 0
is
(a) y=2
(c) y= 2x – 4
(b) y= 2x
(d) y= 2×2 – 4
(2001
12
348 Find differential equation
corresponding to ( y=c(x-c)^{2}, ) where ( c )
is arbitrary constant.
A ( cdot y^{prime 3}=4 yleft(x y^{prime}-2 yright) )
В ( cdot y^{prime 3}=4 yleft(x y^{prime}+2 yright) )
c. ( y^{prime 3}=-8 yleft(x y^{prime}-2 yright) )
D ( cdot y^{prime 2}=-12 yleft(x y^{prime}-2 yright) )
12
349 Obtain a differential equation from the
following equation:
( sin ^{-1} x+sin ^{-1} y=sin ^{-1} c )
12
350 Find the curve passing through the point (0,1) and satisfying the equation ( sin left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)=boldsymbol{a} ) 12
351 The general solution of the differential equation ( frac{d y}{d x}=frac{1-x}{y} ) is a family of curves
which looks most like which of the
following?
( (mathbf{A}) )
(B)
( (mathrm{C}) )
( (mathbf{D}) )
12
352 The order of the differential equation
whose general solution is given by
( boldsymbol{y}=left(boldsymbol{C}_{1}+boldsymbol{C}_{2}right) sin left(boldsymbol{x}+boldsymbol{C}_{3}right)-boldsymbol{C}_{4} boldsymbol{e}^{boldsymbol{x}+boldsymbol{C}_{5}} )
is
A . 5
B. 4
( c cdot 2 )
D. 3
12
353 For each the differential equations given, find the general solution:
( left(1+x^{2}right) d y+2 x y d x=cot x d x(x neq 0) )
12
354 The order of the differential equation of
all parabolas whose axis of symmetry
along ( x ) -axis is:
A .2
B. 3
( c . )
D. None of these
12
355 ( mathbf{f} boldsymbol{y}=cot ^{-1} sqrt{frac{1-sin x}{1+sin x}}, ) find ( frac{d x}{d y} ) 12
356 Solve: ( x^{2} frac{d y}{d x}=frac{y(x+y)}{2} ) 12
357 The solution of the differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=tan left(frac{boldsymbol{y}}{boldsymbol{x}}right)+frac{boldsymbol{y}}{boldsymbol{x}} ) is:
A ( cdot cos left(frac{y}{x}right)=c x )
B. ( sin left(frac{y}{x}right)=c x )
( mathbf{c} cdot cos left(frac{y}{x}right)=c y )
D ( cdot sin left(frac{y}{x}right)=c y )
12
358 Solve the differential equation:
( left(1+e^{2 x}right) d y+left(1+y^{2}right) e^{x} d x=0 )
12
359 The general solution of differentia
equation is ( (y+c)^{2}=c x ) where ( c ) is an
arbitrary constant. The order and degree of the differential equation are respectively:
A .1,2
в. 2,2
c. 1,1
D. 2,1
12
360 Which of the following equation is a linear differential equation of order 3 ?
[Note: The original question asks for linear equation, but it should be linear differential equation]
A ( cdot frac{d^{3} y}{d x^{3}}+frac{d^{2} y}{d x^{2}} frac{d y}{d x}+y=x )
B. ( frac{d^{3} y}{d x^{3}}+frac{d^{2} y}{d x^{2}}+y^{2}=x^{2} )
c. ( quad x frac{d^{3} y}{d x^{3}}+frac{d^{2} y}{d x^{2}}=e^{x} )
D. ( frac{d^{2} y}{d x^{2}}+frac{d y}{d x}=log x )
12
361 Solve the following differential equation:
( boldsymbol{x} cos boldsymbol{y} boldsymbol{d} boldsymbol{y}=left(boldsymbol{x} e^{x} log boldsymbol{x}+boldsymbol{e}^{boldsymbol{x}}right) boldsymbol{d} boldsymbol{x} )
12
362 The general solution of the equation ( frac{d y}{d x}=frac{y^{2}-x}{2 y(x+1)} ) is
( mathbf{A} cdot y^{2}=(1+x) log (1+x)-c )
B. ( _{y^{2}}=(1+x) log frac{c}{(1-x)}-1 )
c. ( y^{2}=(1-x) log frac{c}{(1+x)}-1 )
D. ( y^{2}=(1+x) log frac{c}{(1+x)}-1 )
12
363 The order and degree of the differential equation ( left(boldsymbol{y}^{prime prime prime}right)^{2}+left(boldsymbol{y}^{prime prime}right)^{3}-left(boldsymbol{y}^{prime}right)^{4}+boldsymbol{y}^{5}= )
0 is
A. 3 and 2
B. 1 and 2
( c cdot 2 ) and 3
D. 1 and 4
E. 3 and 5
12
364 ff ( y=x^{x} ), prove that ( frac{d^{2} y}{d x^{2}}-frac{1}{y}left(frac{d y}{d x}right)^{2} )
( frac{boldsymbol{y}}{boldsymbol{x}}=mathbf{0} )
12
365 Find the solution of ( (x+y-1) d y= )
( (boldsymbol{x}+boldsymbol{y}) boldsymbol{d} boldsymbol{x} )
( mathbf{A} cdot 2(y-x)-log (2 x-2 y-1)=k )
B ( cdot 2(y+x)-log (2 x+2 y-1)=k )
( mathbf{c} cdot 2(y-x)-log (x+2 y-1)=k )
D ( cdot 2(y-x)-log (2 x+2 y-1)=k )
12
366 If ( sqrt{frac{boldsymbol{v}}{boldsymbol{mu}}}+sqrt{frac{boldsymbol{mu}}{boldsymbol{v}}}=mathbf{6}, ) then ( frac{boldsymbol{d} boldsymbol{v}}{boldsymbol{d} boldsymbol{mu}}= )
A. ( frac{17 mu-v}{mu-17 v} )
в. ( frac{mu-17 v}{17 mu-v} )
c. ( frac{17 mu+v}{mu-17 v} )
D. ( frac{mu+17 v}{17 mu-v} )
12
367 Solve the differential equation: ( frac{d y}{d x}- ) ( x sin ^{2} x=frac{1}{x log x} ) 12
368 Find the general solution of the differential equation ( e^{x} tan y d x+left(1-e^{x}right) sec ^{2} y d y=0 ) 12
369 Find the deff equation ( left(x^{2}+y^{2}right) d x )
( 2 x y d y=0 )
12
370 Find general solution of ( y-x frac{d y}{d x}= )
( bleft(1+x^{2} frac{d y}{d x}right) ) is:
( begin{array}{ll}text { A. } & b+k x=y(1+b x)end{array} )
В. ( quad b+k y=x(1+b x) )
( begin{array}{ll}text { c. } & b+k y=x(1+b y)end{array} )
D. ( quad b+k x=x(1+b y) )
12
371 Solution of ( sin ^{-1}left[frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right]=boldsymbol{x}+boldsymbol{y} ) is:
A ( cdot 1+tan left(frac{x+y}{2}right)=-frac{2}{x+c} )
B. ( 1+cos left(frac{x+y}{2}right)=-frac{2}{x+c} )
c. ( 1+sec left(frac{x+y}{2}right)=-frac{2}{x+c} )
D. ( 1+cos left(frac{x+y}{2}right)=frac{2}{x+c} )
12
372 The solution of ( frac{d y}{d x}=frac{sqrt{x^{2}-y^{2}}+y}{x} ) 12
373 Determine the order and degree(if defined) of the following differentia
equation. ( left(frac{boldsymbol{d} boldsymbol{s}}{boldsymbol{d} boldsymbol{t}}right)^{boldsymbol{4}}+boldsymbol{3} boldsymbol{s} frac{boldsymbol{d}^{2} boldsymbol{s}}{boldsymbol{d} boldsymbol{t}^{2}}=mathbf{0} )
12
374 ( int_{0}^{pi / 2} sin x cos x d x ) is equal to: 12
375 Match the elements of list I, which have differential equations, with elements of list II, which have solutions of
differential equations:
List I
[
text { A) } y y_{1}=sec ^{2} x
]
1) ( y sec ^{2} x=sec x+c )
В) ( y_{1}=x sec y )
2) ( x y=cos y+c )
C) ( y_{1}+(2 y tan x)=sin x )
3) ( x y=sin x+c )
D) ( x y_{1}+y=cos x )
[
text { 4) } y^{2}=2 tan x+c
]
[
text { 5) } x^{2}=2 sin y+c
]
Then, the correct order for ( boldsymbol{A} boldsymbol{B} boldsymbol{C} boldsymbol{D} ) is:
( mathbf{A} cdot 3254 )
B. 4123
c. 4513
D. 3512
12
376 Solution of ( frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{d} boldsymbol{y}}+boldsymbol{m} boldsymbol{x}=mathbf{0}, ) where ( boldsymbol{m}<mathbf{0} )
is:
A ( . x=C e^{m y} )
B . ( x=C e^{-m y} y )
c. ( x=m y+C )
D. ( x=C )
12
377 Consider the family of all circles whose centers lie on the
straight line y = x. If this family of circle is represented by
the differential equation Py” + O’+1= 0, where P, Q are
dy
dy!
functions of x, y and y’ heres
2
, then
which of the following statements is (are) true?
(JEE Adv. 2015)
(a) P=y+x
(b) P=y- x
(c) P+Q=1-x+y+y+692 (d) P-Q=x+y-y-032
12
378 Find the degree of homogeneity of function ( boldsymbol{f}(boldsymbol{x}, boldsymbol{y})=boldsymbol{a} boldsymbol{x}^{2 / 3}+boldsymbol{h} boldsymbol{x}^{1 / 3} boldsymbol{y}^{1 / 3}+ )
( b y^{2 / 3} )
( A cdot 2 / 3 )
B. 2
( c cdot 3 )
D. 3/2
12
379 Form the differential equation
corresponding to ( y^{2}-2 a y+x^{2}=a^{2} b y )
eliminating a.
12
380 If ( boldsymbol{d} boldsymbol{x}+boldsymbol{d} boldsymbol{y}=(boldsymbol{x}+boldsymbol{y})(boldsymbol{d} boldsymbol{x}-boldsymbol{d} boldsymbol{y}) Rightarrow boldsymbol{y}+ )
( log (x+y)=x+C ) is the solution of
differential equation
12
381 Let ( p in I R ) then the differential
equation of the family of curves ( y= )
( (alpha+beta x) e^{p x}, ) where ( alpha, beta ) are arbitrary
constants, is:
A ( cdot y^{prime prime}+4 p y^{prime}+p^{2} y=0 )
в. ( y^{prime prime}-2 p y^{prime}+p^{2} y=0 )
c. ( y^{prime prime}+2 p y^{prime}-p^{2} y=0 )
D. ( y^{prime prime}+2 p y^{prime}+p^{2} y=0 )
12
382 In each of the exercises 1 to 3 , form a
differential equation representing the given family of curves by eliminating arbitrary constants ( a ) and ( b )
1. ( frac{x}{a}+frac{y}{b}=1 )
2. ( y^{2}=aleft(b^{2}-x^{2}right) )
3. ( boldsymbol{y}=boldsymbol{a} e^{3 x}+boldsymbol{b} e^{2 x} )
12
383 The differential equation representing the family of curves given by ( boldsymbol{y}= )
( boldsymbol{a} e^{-boldsymbol{3} x}+boldsymbol{b}, ) where ( boldsymbol{a} ) and ( boldsymbol{b} ) are arbitrary
constants, is :
( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+boldsymbol{3} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}-boldsymbol{2} boldsymbol{y}=mathbf{0} )
A ( cdot frac{d^{2} y}{d x^{2}}+3 frac{d y}{d x}-2 y=0 )
B. ( frac{d^{2} y}{d x^{2}}-3 frac{d y}{d x}=0 )
c. ( frac{d^{2} y}{d x^{2}}-3 frac{d y}{d x}-2 y=0 )
D ( cdot frac{d^{2} y}{d x^{2}}+3 frac{d y}{d x}+2 y=0 )
E ( cdot frac{d^{2} y}{d x^{2}}+3 frac{d y}{d x}=0 )
12
384 ( boldsymbol{y}=boldsymbol{x}^{2}+boldsymbol{3} boldsymbol{x}+2 ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
385 The D.E of the family of prabolas of which has a latus retum and whose
axes are parallel to ( x- ) axis
A ( cdot y_{1}^{3}+2 a y_{2}=0 )
B . ( y_{1}^{3}+a y_{2}=0 )
c. ( y_{1}^{3}+4 a y_{2}=0 )
D. ( y_{1}^{3}+3 a y_{2}=0 )
12
386 ( left(x^{2} y-2 x y^{2}right) d x=left(x^{3}-3 x^{2} yright) d y )
A ( cdot k y^{3} e^{x / y}=x^{2} )
B. ( k x^{3} e^{x / y}=x^{2} )
c. ( k x^{3} e^{x / y}=y^{2} )
D. ( k y^{3} e^{x / y}=y^{2} )
12
387 Find the differential equation
representing the family of curves ( boldsymbol{v}= ) ( frac{boldsymbol{A}}{boldsymbol{r}}+boldsymbol{B}, ) where ( boldsymbol{A} ) and ( boldsymbol{B} ) are arbitary
constants.
12
388 Solve the following differential equation:
( left(y-x y^{2}right) d x-left(x+x^{2} yright) d y=0 )
A ( cdot x=c y e^{x y} )
В. ( y=c x e^{x y} )
c. ( y=-c x e^{x y} )
D. ( x=c y e^{-x y} )
12
389 20.
If a curve y = f(x) passes through the point (1, -1) ana
satisfies the differential equation, y(1 + xy) dx = x dy, th
f(3) is equal to :
is equal to :
[JEE M 2016)
12
390 Solve ( : boldsymbol{x}^{2} boldsymbol{y} boldsymbol{d} boldsymbol{x}=left(boldsymbol{x}^{3}+boldsymbol{y}^{3}right) boldsymbol{d} boldsymbol{y}=mathbf{0} ) 12
391 Solution of differential equation ( (x cos x-sin x) d x=frac{x}{y} sin x d y ) is
( mathbf{A} cdot sin x=ln |x y|+c )
B. ( ln left|frac{sin x}{x}right|=y+c )
( c cdotleft|frac{sin x}{x y}right|=c )
D. None of these
12
392 Solution of ( left(frac{boldsymbol{x}+boldsymbol{y}-boldsymbol{a}}{boldsymbol{x}+boldsymbol{y}-boldsymbol{b}}right)left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)= )
( left(frac{boldsymbol{x}+boldsymbol{y}+boldsymbol{a}}{boldsymbol{x}+boldsymbol{y}+boldsymbol{b}}right) )
A ( cdot log left[(x+y)^{2}-a bright]=frac{2}{b-a}[x-y]+k )
B ( cdot log left[(x+y)^{2}+a bright]=frac{1}{b-a}[x-y]+k )
c. ( left(frac{b-a}{2}right)left[log left((x+y)^{2}-a bright)right]=x+c )
D.
12
393 ( frac{d^{2} y}{d x^{2}}+3left(frac{d y}{d x}right)^{2}=x^{2} log left(frac{d^{2} y}{d x^{2}}right) ) 12
394 Obtain a differential equation by
eliminating ( c ) when it is given
( tan ^{-1} x+tan ^{-1} y=tan ^{-1} c )
12
395 Find the differential equation of the family of all striaght lines passing through the origin. 12
396 If ( boldsymbol{y}^{prime}-mathbf{3} boldsymbol{y}^{prime}+mathbf{2} boldsymbol{y}=mathbf{0} ) where ( boldsymbol{y}(mathbf{0})= )
1 ( , y^{prime}(0)=0, ) then the value of ( y ) at ( x= )
( log 2 ) is
A .
B. – –
( c cdot 2 )
D.
12
397 Find the general solution of the differential equation:
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{y}}{boldsymbol{x}}+sin left(frac{boldsymbol{y}}{boldsymbol{x}}right) )
12
398 A ladder ( 24 mathrm{ft} ) long leans against a vertical wall. The lower end is moving away at the rate of ( 3 mathrm{ft} ) lsec. Find the rate
at which the top of the ladder is moving downwards, if its foot is ( 8 mathrm{ft} ) from the
wall.
12
399 Find the order of the differential
equations of all circles in the plane XOY
which have their centres on x-axis and
have give radius.
12
400 The solution of the differential equation ( frac{d y}{d x}+frac{2 y x}{1+x^{2}}=frac{1}{left(1+x^{2}right)^{2}} )
A ( cdot yleft(1+x^{2}right)=c+tan ^{-1} x )
B. ( frac{y}{1+x^{2}}=c+tan ^{-1} x )
C・ ( y log left(1+x^{2}right)=c+tan ^{-1} x )
D ( cdot yleft(1+x^{2}right)=c+sin ^{-1} x )
12
401 The general solution of the differential equation, ( sin 2 xleft(frac{d y}{d x}-sqrt{tan x}right)-y= )
( 0, ) is
( mathbf{A} cdot y sqrt{tan x}=x+c )
B. ( y sqrt{cot x}=tan x+c )
c. ( y sqrt{tan x}=cot x+c )
D. ( y sqrt{cot x}=x+c )
12
402 11. If length of tangent at any point on the curve y = f(x)
intecepted between the point and the x-axis is of length 1.
Find the equation of the curve. (2005 – 4 Marks)
12
403 Let ( mathbf{y}(mathbf{x}) ) be the solution of the
differential equation ( (mathbf{x} log mathbf{x}) frac{mathbf{d y}}{mathbf{d x}}+mathbf{y}=2 mathbf{x} log mathbf{x},(mathbf{x} geqslant 1) )
Then ( mathbf{y}(mathbf{e}) ) is equal to :
A.
B. 2
( c cdot z )
D.
12
404 If ( y=sqrt{x}^{sqrt{x} sqrt{x}^{sqrt{x}} ldots infty} ) then show that
( x frac{d y}{d x}=frac{y^{2}}{2-y log x} )
12
405 Family ( y=a x+a^{3} ) of curve
repersented by the differential equation of degree
( A ). three
B. two
c. one
D. four
12
406 The order and degree of the differential equation of the family of circles of fixed
radius ( r ) with centres on the ( y ) -axis, are
respectively.
A .2,2
в. 2,3
( mathrm{c} cdot 1, )
D. 3,1
E. 1,2
12
407 If the length of the tangent at any point on the curve ( y=f(x) ) intercepted
between the point of contact and ( x ) -axis
is of length ( 1, ) the equation of the curve
¡s:
This question has multiple correct options
( mathbf{A} cdot sqrt{1-y^{2}}+ln |(1-sqrt{1-y^{2}}) / y|=pm x+c )
B ( cdot sqrt{1-y^{2}}-ln |(1-sqrt{1-y^{2}}) / y|=pm x+c )
( mathbf{c} cdot sqrt{1-y^{2}}+ln |(1+sqrt{1-y^{2}}) / y|=pm x+c )
D. None of these
12
408 Find the differential equation corresponding to the family of curves
( y=k(x-k)^{2} ) where ( k ) is an arbitrary
constant.
( ^{mathbf{A}} cdotleft(frac{d y}{d x}right)^{3}-4 x y frac{d y}{d x}+8 y^{2}=0 )
( ^{mathrm{B}}left(frac{d y}{d x}right)^{3}+4 x y frac{d y}{d x}+8 y^{2}=0 )
( ^{mathrm{c}}left(frac{d y}{d x}right)^{3}-2 x y frac{d y}{d x}+8 y^{2}=0 )
( ^{mathrm{D} cdot}left(frac{d y}{d x}right)^{3}+2 x y frac{d y}{d x}+8 y^{2}=0 )
12
409 The solution of the differential equation ( frac{d^{2} y}{d x^{2}}+3 y=-2 x ) is
A ( cdot c_{1} cos sqrt{3} x+c_{2} sin sqrt{3} x-frac{2}{3} x )
B. ( c_{1} cos sqrt{3} x+c_{2} sin sqrt{3} x-frac{4}{5} )
( mathbf{c} cdot_{c_{1} cos sqrt{3} x+c_{2} sin sqrt{3} x-2 x^{2}+frac{4}{9}} )
D. ( c_{1} cos sqrt{3} x+c_{2} sin sqrt{3} x-frac{2}{3} x^{2}+frac{4}{9} )
12
410 If ( boldsymbol{2} boldsymbol{x}=boldsymbol{y}^{frac{1}{5}}+boldsymbol{y}^{-frac{1}{5}} ) and ( left(boldsymbol{x}^{2}-mathbf{1}right) frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+ )
( lambda x frac{d y}{d x}+k y=0, ) then ( lambda+k ) is equal to:
A . -23
в. -24
( c cdot 26 )
D. -26
12
411 The differential equation of a body projected from the earth is given by ( frac{d v}{d t}=-g-k v . ) The distance travelled by the body at any time ( t ) is given by ( x= ) ( left(frac{boldsymbol{g}}{boldsymbol{k}^{2}}+frac{boldsymbol{u}}{boldsymbol{k}}right)left(mathbf{1}+boldsymbol{e}^{-boldsymbol{k} boldsymbol{t}}right)-frac{boldsymbol{g}}{boldsymbol{k}} boldsymbol{t} )
A. True
B. False
12
412 Solve the differential equation:
( (x-sin y) d y+tan y d x=0 )
12
413 The solution of the differential equation ( sin y frac{d y}{d x}=cos y(1-x cos y) ) is:
A ( cdot sec y=x+1+K e^{-x} )
B. sec ( y=x+1+K e^{x} )
c. sec ( y=x-1+K e^{x} )
D. sec ( y=x-1+K e^{-x} )
12
414 The differential equation of family of coaxial circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{2} boldsymbol{lambda} boldsymbol{x}+boldsymbol{c}=boldsymbol{0} ) is
A ( cdot frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}+1=0 )
B. ( y cdot frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}=0 )
( ^{mathbf{c}}left(frac{d^{2} y}{d x^{2}}right)+left(frac{d y}{d x}right)^{2}+1=0 )
D ( frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}=0 )
12
415 Solve: ( boldsymbol{x} boldsymbol{d} boldsymbol{y}=left(boldsymbol{y}+boldsymbol{x} cos ^{2}left(frac{boldsymbol{y}}{boldsymbol{x}}right)right) boldsymbol{d} boldsymbol{x} ) 12
416 Solve ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x} cos boldsymbol{x} ) 12
417 If the curve ( boldsymbol{y}=c_{1} e^{m_{1} x}+c_{2} e^{m_{3} x}, ) where
( c_{1}, c_{2}, c_{3} ) are arbitrary constants and
( m_{1}, m_{2}, m_{3} ) are roots of ( m^{3}-7 m+ )
( mathbf{6}=mathbf{0}, ) then the differential equation corresponding to the given curve is,
A ( cdot y_{2}-7 y_{1}+6 y=0 )
в. ( y_{2}-7 y_{1}+6=0 )
c. ( y_{2}+7 y_{1}+6 y=0 )
D. ( y_{2}+7 y_{1}+6=0 )
12
418 Form the differential equation from the following primitives, where constant is arbitrary. ( boldsymbol{y}=boldsymbol{c} boldsymbol{x}+boldsymbol{2} boldsymbol{c}^{2}+boldsymbol{c}^{boldsymbol{3}} ) 12
419 If ( y=e^{m sin ^{-1} x}, ) then show that ( (1- )
( left.x^{2}right) frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}-m^{2} y=0 )
12
420 The differential equation of all
parabola’s with axis parallel to axis of ( y )
-axis is:
A ( cdot y_{2}=2 y_{1}+x )
B ( cdot y_{3}=2 y_{1} )
c. ( y_{2}^{3}=y_{1} )
D . ( y_{3}=0 )
12
421 The slope of the tangent at ( (x, y) ) of ( a ) curve passing through the point (2,2) is ( frac{x}{y}, ) then the equation of the curve is
A ( cdot x^{2}+y^{2}=4 )
B . ( x^{2}-y^{2}=4 )
c. ( x^{2}=y^{2} )
D. ( x^{2}=y^{2}-4 )
12
422 Solve:
( frac{d y}{d x}=2 x^{2}+x ; ) find ( y ) when ( x=0 )
12
423 Find the solution of ( frac{d y}{d x}+frac{y}{x}= )
( frac{1}{(1+log x+log y)^{2}} )
A ( cdot x yleft[1-(log x y)^{2}right]=-frac{x^{2}}{2}+C )
B. ( x yleft[1+(log x y)^{2}right]=frac{x^{2}}{2}+C )
c. ( x yleft[1+(log x y)^{2}right]=-frac{x^{2}}{2}+C )
D. ( x yleft[1-(log x y)^{2}right]=frac{x^{2}}{2}+C )
12
424 Represent the following families of curves by forming the corresponding differential equation.(a, b being parameters). ( x^{2}-y^{2}=a^{2} ) 12
425 The degree and order of ( (2 x+ )
( mathbf{3} boldsymbol{y})left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{3}+mathbf{7} boldsymbol{x} frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+boldsymbol{y}=boldsymbol{e}^{2 boldsymbol{x}} ) are:
A .2,2
B. 1,3
( c cdot 1,2 )
D. 2,
12
426 If ( mathbf{Y}=(x+sqrt{x^{2}-1})^{m} ) show that
( left(x^{2}-1right) frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}=m^{2} y )
12
427 Form the differential equation
representing the family of curves ( boldsymbol{y}= )
( boldsymbol{m} boldsymbol{x}, ) where ( m ) is arbitrary constant.
12
428 Differential equation having
( mathbf{y}=left(sin ^{-1} mathbf{x}right)^{2}+mathbf{A}left(cos ^{-1} mathbf{x}right)+mathbf{B} ) where
A, B are arbitrary constants as general solution is
A ( cdotleft(1-x^{2}right) y_{2}-x y_{1}=2 )
B ( cdotleft(1-x^{2}right) y_{2}-x y_{1}=1 )
c. ( (1-x) y_{2}-x y_{1}=2 )
D. ( left(1-mathrm{x}^{2}right) mathrm{y}_{2}-mathrm{xy}_{1}=4 )
12
429 The order and degree of the differentia
equation ( left[1+left(frac{d y}{d x}right)^{2}+sin left(frac{d y}{d x}right)right]^{3 / 4}=frac{d^{2} y}{d x^{2}} )
are:
A. order ( =2 ; ) degree ( =3 )
B. order= ( 2 ; ) degree ( =4 )
( ^{mathrm{c}} cdot_{mathrm{order}=} 2 ; ) degree ( =frac{3}{4} )
D. order= ( 2 ; ) degree ( = ) not defined
12
430 Form the differential equation of the family of curves represented by the equation(a being the parameter). ( (x-a)^{2}+2 y^{2}=a^{2} ) 12
431 What is the half-life of a Radium-226 if
its decay rate is ( 0.000436 ? )
A ( . t=1237 )
B . ( t=1365 )
c. ( t=1440 )
D. ( t=1590 )
12
432 If ( y=a cos (log x)+b sin (log x), ) then
A. ( x^{2} frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}+y=0 )
B. ( quad x^{2} frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}-y=0 )
c. ( x^{2} frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}-y=0 )
D. ( x^{2} frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}+x y=0 )
12
433 Solve
( int frac{d y}{y}=int frac{d x}{x} )
12
434 The differential equation of all parabolas having their axis of symmetry with the axis of ( x ) is?
A ( cdot quad y frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}=0 )
B. ( quad y frac{d^{2} x}{d x^{2}}+left(frac{d y}{d x}right)^{2}=0 )
( mathbf{c} cdot y=2 x frac{d y}{d x} )
D. ( y frac{d^{2} y}{d x^{2}}-left(frac{d y}{d x}right)=0 )
12
435 Find the general solution of each of the following differential equations:
( left(e^{x}+e^{-x}right) d y-left(e^{x}-e^{-x}right) d x=0 )
12
436 The differential equation by eliminating ( A ) and ( B ) from ( y=A x^{3}+B x^{2} ) is
( ^{mathbf{A}} cdot_{x^{2}}left(frac{d^{2} y}{d x^{2}}right)+4 xleft(frac{d y}{d x}right)+6 y=0 )
в. ( x^{2}left(frac{d^{2} y}{d x^{2}}right)-4 xleft(frac{d y}{d x}right)-6 y=0 )
c. ( x^{2}left(frac{d^{2} y}{d x^{2}}right)-4 xleft(frac{d y}{d x}right)+6 y=0 )
D ( x^{2}left(frac{d^{2} y}{d x^{2}}right)-5 xleft(frac{d y}{d x}right)+6 y=0 )
12
437 The degree of the differential equation ( frac{d y}{d x}-x=left(y-x frac{d y}{d x}right)^{-4} )
( A cdot 2 )
B. 3
( c cdot 4 )
( D )
12
438 If ( left(1+x^{2}right) frac{d y}{d x}=x(1-y), y(0)=frac{4}{3} )
then the value of ( y(sqrt{8})+frac{8}{9} i s )
( A cdot 4 )
B. 2
( c cdot 3 )
D.
12
439 The solution of the equation ( frac{d y}{d x}+ )
( boldsymbol{x}(mathbf{2} boldsymbol{x}+boldsymbol{y})=boldsymbol{x}^{3}(boldsymbol{2} boldsymbol{x}+boldsymbol{y})^{3}-boldsymbol{2} ) is ( (C )
being arbitrary constant):
A ( cdot frac{1}{2 x+x y}=x^{2}+1+C e^{x} )
B. ( frac{1}{(2 x+y)^{2}}=x^{2}+1+C e^{x^{2}} )
c. ( frac{1}{2 x+y}=x+1+C e^{-x^{2}} )
D. ( frac{1}{(2 x+y)^{2}}=x^{2}+1+C )
12
440 Solve the following differential equation ( cos x frac{d y}{d x}-cos 2 x=cos 3 x ) 12
441 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: boldsymbol{y}=cos ^{-1}left(frac{boldsymbol{2} boldsymbol{x}}{boldsymbol{1}+boldsymbol{x}^{2}}right),-boldsymbol{1}< )
( boldsymbol{x}<mathbf{1} )
12
442 The differential equation whose solution is ( boldsymbol{y}=boldsymbol{a} cos (boldsymbol{3} boldsymbol{x}+boldsymbol{b}) ) is?
A ( cdot y_{2}+3 y=0 )
в. ( y_{2}+y=0 )
c. ( y_{2}+9 y=0 )
D. ( y_{2}+6 y=0 )
12
443 The envelope of a family of curves is a
curve ( f(x, y, c)=0 ) whose equation is
obtained by eliminating the parameter ( c ) from ( f(x, y, c)=0 ) and ( frac{partial f}{partial c}=0 )
where ( frac{partial f}{partial c} ) is the differential coefficient of f with respect to c, treating x and y as constants. Moreover, the envelope of the family of normals to a curve is known as the evolute of the curve. The envelope of the family of straight lines whose sum of intercepts on the axes is 4 is:
A ( cdot sqrt{x}+sqrt{y}=2 )
в. ( (x-y)^{2}-8(x+y)+16=0 )
c. ( (x-y)^{2}=4(x+y) )
D. ( (x+y)^{2}=4(x-y) )
12
444 Show that the given differential equation is homogenous and solve them ( boldsymbol{Y}^{prime}=frac{boldsymbol{x}+boldsymbol{y}}{boldsymbol{x}} ) 12
445 The solution of the differential equation, ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=(boldsymbol{x}-boldsymbol{y})^{2}, ) when ( boldsymbol{y}(mathbf{1})=mathbf{1}, ) is
( ^{mathrm{A}} cdot log _{e}left|frac{2-y}{2-x}right|=2(y-1) )
( ^{mathbf{B}} cdot log _{e}left|frac{2-x}{2-y}right|=x-y )
c. ( -log _{e}left|frac{1+x-y}{1-x+y}right|=x+y-2 )
D. ( -log _{e}left|frac{1-x+y}{1+x-y}right|=2(x-1) )
12
446 The number of days so that half the population have flu is
( mathbf{A} cdot 50 )
B. 40
c. 45
D. 25
12
447 The differential equation for all the straight lines which are at a unit distance from the origin is
( ^{A} cdotleft(y-x frac{d y}{d x}right)^{2}=1-left(frac{d y}{d x}right)^{2} )
( ^{mathrm{B}}left(y+x frac{d y}{d x}right)^{2}=1+left(frac{d y}{d x}right)^{2} )
( ^{mathrm{c}}left(y-x frac{d y}{d x}right)^{2}=1+left(frac{d y}{d x}right)^{2} )
( ^{mathrm{D} cdot}left(y+x frac{d y}{d x}right)^{2}=1-left(frac{d y}{d x}right)^{2} )
12
448 Form the differential equation by eliminating arbitrary constants from the relation ( boldsymbol{y}=boldsymbol{A} boldsymbol{e}^{boldsymbol{5} boldsymbol{x}}+boldsymbol{B} boldsymbol{e}^{-boldsymbol{5} boldsymbol{x}} ) 12
449 The general solution of differential
equation is ( y=a e^{b x+c} ) where ( a, b, c ) are
arbitrary constant. The order of differential equation is:
( A cdot 3 )
B . 2
( c cdot 1 )
D. none of these
12
450 A particle is in motion along a curve ( 12 y-x^{3} . ) The rate of change of its
ordinate exceeds that of abscissa in
This question has multiple correct options
A ( .-2<x<2 )
B. ( x=pm 2 )
c. ( x2 )
12
451 A curve ( y=f(x) ) which passes through
(4, 0) satisfies the differential equation ( x d y+2 y d x=x(x-3) d x . ) The area
bounded by ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) and line ( boldsymbol{y}=boldsymbol{x} ) (in
square unit) is
A . 32
B. ( frac{64}{3} )
c. ( frac{128}{3} )
D. 64
12
452 Find a particular solution of the differential equation ( frac{d y}{d x}+y cot x= ) ( 4 x operatorname{cosec} x(x neq 0), ) given that ( y=0 )
when ( boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{2}} )
12
453 Solve: ( frac{d y}{d x}+x frac{y}{x}=frac{y^{2}}{x^{2}} ) 12
454 Solve the differential equation, ( frac{d y}{d x}- ) ( frac{2 x y}{1+x^{2}}=x^{2}+1 ) 12
455 ( 2 x y frac{d y}{d x}=x^{2}+y^{2} ) 12
456 The solution of differential equation ( left(x^{2}+y^{2}right) d y=x y d x ) is ( y=y(x) . ) If
( boldsymbol{y}(mathbf{1})=mathbf{1} ) and ( boldsymbol{y}left(boldsymbol{x}_{0}right)=boldsymbol{e}, ) then ( boldsymbol{x}_{0} ) is:
A ( cdot sqrt{2left(mathrm{e}^{2}-1right)} )
B . ( sqrt{2left(mathrm{e}^{2}+1right)} )
( c cdot sqrt{3} e )
D. ( sqrt{frac{e^{2}+1}{2}} )
12
457 If ( boldsymbol{y}=[log (boldsymbol{x}+sqrt{boldsymbol{x}^{2}+mathbf{1}})]^{2} )
Prove that ( left(mathbf{1}+boldsymbol{x}^{2}right) frac{d^{2} boldsymbol{y}}{d x^{2}}+boldsymbol{x} frac{d boldsymbol{y}}{d boldsymbol{x}}=mathbf{2} )
12
458 The particular solution of ( frac{d y}{d x}+ ) ( frac{boldsymbol{y}+boldsymbol{2}}{boldsymbol{x}+boldsymbol{2}}=boldsymbol{0} ) at (1,2) is given by
A ( . x y+2(x+y)=8 )
B. ( x y+8(x+y)=2 )
c. ( (x+2)+2(y+2)=8 )
D. ( 8(x+2)+(y+2)=2 )
12
459 JU
18.
(0) 00
Let the population of rabbits surviving at time t be governed
.
dt
by the differential equation ==p(t)– 200. If
p(O)= 100, then p(t) equals:
[JEE M 2014]
(a) 600 – 500 et/2 (b) 400 – 300 e =+/2.
(C) 400-300 et/2 (d) 300 – 200 e 1/2
12
460 Show that the curve such that the
distance between the origin and the tangent at an arbitrary point is equalto the distance between the origin and the normal at the same point, ( sqrt{x^{2}+y^{2}}= ) ( boldsymbol{c} e^{pm tan ^{-1} frac{y}{x}} )
12
461 Solve: ( frac{d y}{d x}+2 y=sin x ) 12
462 Solve the differential equation ( y e^{frac{x}{y}} d x=left(x e^{frac{x}{y}}+y^{2}right) d y(y neq 0) ) 12
463 Form the differential equation from the following primitive, where constant is arbitrary. ( boldsymbol{x} boldsymbol{y}=boldsymbol{a}^{2} ) 12
464 The order, degree of the ( D . E )
corresponding to the family of curve
( boldsymbol{y}=boldsymbol{a}(boldsymbol{x}+boldsymbol{a})^{2} ) where ( boldsymbol{a} ) is an arbitrary
constant is
A . 1,2
в. 2,4
( c .3,1 )
D. 1,3
12
465 ( A(0,0) ) and ( B(8,2) ) are fixed points on the curve ( y^{3}=|x| . ) A point ( C ) starts
moving from origin along the curve for ( x<0 ) such that rate of change in the
ordinate is ( 2 mathrm{cm} / mathrm{s} ). After ( t_{0} ) seconds
triangle ( A B C ) become a right
triangle. After ( t_{0} ) seconds, tangent is
drawn to the curve at point ( C ) to
intersect it again at ( (boldsymbol{alpha}, boldsymbol{beta}) ) then ( mathbf{4} boldsymbol{alpha}+ )
( mathbf{3} boldsymbol{beta} ) is
A ( cdot frac{4}{3} )
B. ( frac{3}{4} )
( c cdot 2 )
( D )
12
466 14.
If
7 = y +3> 0 and y (0)=2, then y (In 2) is equal to :
20
(a) 5
(c) – 2
(b) 13
(d) 7
12
467 Integrating factor of ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+frac{mathbf{1}}{boldsymbol{x} log boldsymbol{x}} boldsymbol{y}= )
( frac{2}{x^{2}} ) is:
A ( cdot e^{x} )
B. ( log x )
( c cdot frac{1}{x} )
D ( cdot e^{-x} )
12
468 The D.E of the family of ellipse with centre at the origin and having co ordinate axes as axes is
A ( . xleft[y y_{2}+y_{1}^{2}right]=y y_{1} )
B . ( x^{2} y_{2}+y_{1}^{2}=y )
( mathbf{c} cdot x y y_{2}+y_{1}^{2}=y )
D. ( x y y_{2}+y_{1}^{2}+y=0 )
12
469 The differential equation of the family of circles with fixed radius 5 units and
centre on the line ( y=2 ) is
A ( cdot(x-2)left(y^{1}right)^{2}=25-(y-2)^{2} )
B . ( (y-2)left(y^{1}right)^{2}=25+(y-2)^{2} )
C ( cdot(y-2)^{2}left(y^{1}right)^{2}=25-(y-2)^{2} )
D. ( (x-2)^{2}left(y^{1}right)^{2}=25-(y-2)^{2} )
12
470 The D.E of the family of circles which
touch the ( y ) -axis at the origin is:
A ( y_{1}=frac{x^{2}-y^{2}}{x y} )
B. ( y_{1}=frac{x^{2}-y^{2}}{2 x y} )
C ( y_{1}=frac{y^{2}-x^{2}}{2 x y} )
D. ( y_{1}-left(y^{2}-x^{2}right)+x y=0 )
12
471 The order of the differential equation, of
which ( x y=c e^{x}+b e^{-x}+x^{2} ) is a
solution, is:
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot 3 )
D. none of these
12
472 Obtain a differential equation by eliminating the arbitrary constants ( A, B ) from the following equation. ( boldsymbol{y}=boldsymbol{A} cos boldsymbol{t}-boldsymbol{B} sin boldsymbol{t} ) 12
473 Find the curve for which the sum of the
lengths of the tangent and subtangent at any of its point is proportional to the product of the coordinates of the point of tangency, the proportionality factor is equal to k.
A ( cdot y=frac{1}{k} ln left|cleft(k^{2} x^{2}-1right)right| )
B. ( y=frac{1}{k} ln mid cleft(k x^{2}-1right) )
c. ( y=frac{1}{k} ln mid cleft(k x^{2}+1right) )
D. ( y=frac{1}{k} ln mid cleft(k^{2} x^{2}+1right) )
12
474 The solution of the differential equation ( x frac{d y}{d x}=frac{y}{1+log x} ) is :
A. ( y=log x+C )
в. ( y=frac{C}{1+log x} )
c. ( y=C(x+log x) )
D. ( y=x+log (C x) )
E . ( y=C(1+log x) )
12
475 The solution of ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{y}}{boldsymbol{x}}+sin left(frac{boldsymbol{y}}{boldsymbol{x}}right) ) is 12
476 The order of the differential equation of the parabola whose axis is parallel to ( y- ) axis is:
( A )
B. 2
( c cdot 3 )
D. 4
12
477 If the differential equation of a body falling from rest under gravity is given by ( v frac{d v}{d x}+frac{n^{2}}{g} v^{2}=g, ) then the velocity of
the body is given by ( v^{2}=frac{g^{2}}{n^{2}}(1- )
( left.e^{-2 n^{2} x / g}right) )
A. True
B. False
12
478 To change ( (3 x+4 y+5)-(2 x+3 y+ )
4) ( frac{d y}{d x}=0 ) into homogeneous equation
origin is shifted to ( (boldsymbol{h}, boldsymbol{k}) ) then ( boldsymbol{h}+boldsymbol{k}= )
( A cdot 3 )
B.
c. -2
D. –
12
479 If the differential equation of all straight
lines which are at a fixed distance of 10
units from origin is
( left(y-x y_{1}right)^{2}=Aleft(1+y_{1}^{2}right) ) then ( frac{A}{100} ) is
equal to ( ldots ) units.
12
480 The solution of differential equation ( frac{d y}{d x}+y tan x=2 x+x^{2} tan x ) is?
A ( cdot y=x^{2}+c cos x )
B . ( y=2 x^{2}-c cos x )
C ( cdot y+x^{2}=c cos x )
D. ( y+2 x^{2}=c cos x )
12
481 ff ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{a} boldsymbol{x}} ) cos ( boldsymbol{b} boldsymbol{x}, ) then find the value of
( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}-boldsymbol{2} boldsymbol{a} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+left(boldsymbol{a}^{2}+boldsymbol{b}^{2}right) boldsymbol{y} )
A . 0
B. 1
( mathbf{c} cdot e^{a x} )
D. 3a
12
482 4.
Determine the equation of the curve passing through the
origin, in the form y=f(x), which satisfies the differential
dy
equation
= sin (10x + 6y).
(1996 – 5 Marks)
12
483 The general solution of the differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{e}^{boldsymbol{y}+boldsymbol{x}}+boldsymbol{e}^{boldsymbol{y}-boldsymbol{x}} ) is
A ( cdot e^{-y}=e^{x}-e^{-x}+c )
B . ( e^{-y}=e^{-x}-e^{x}+c )
( mathbf{c} cdot e^{-y}=e^{x}+e^{-x}+c )
D. ( e^{y}=e^{x}+e^{-x}+c )
where ( c ) is an arbitrary constant
12
484 Consider the following statements:
1. The general solution of ( frac{d y}{d x}=f(x)+ ) ( x ) is of the form ( y=g(x)+c, ) where ( c ) is
an arbitrary constant.
2. The degree of ( left(frac{d y}{d x}right)^{2}=f(x) ) is 2 Which of the above statements is/are
correct?
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
485 The General solution for the equation ( (2 sqrt{x y}-x) d y+y d x=0 ) is?
A ( cdot y=c e^{-sqrt{x / y} / y} )
B ( cdot y=c e^{sqrt{y / x}} )
C ( cdot x=c e^{sqrt{y / x}} )
D. None of these
12
486 Form the differential equation from the following primitive, where constant is arbitrary. ( boldsymbol{x} boldsymbol{y}=boldsymbol{a}^{2} ) 12
487 Find the general solution of the differential equation:
( boldsymbol{x} boldsymbol{d} boldsymbol{y}-boldsymbol{y} boldsymbol{d} boldsymbol{x}=sqrt{boldsymbol{x}^{2}+boldsymbol{y}^{2}} boldsymbol{d} boldsymbol{x} )
12
488 Find order and degree (if defined) of the differential equation ( frac{boldsymbol{d}^{4} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{4}}+ )
( sin left(frac{boldsymbol{d}^{3} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{3}}right)=mathbf{0} )
12
489 10.
u U lu Uluer anu birst degree
The differential equation of all circles passing through the
origin and having their centres on the x-axis is [2007]
(a) y2 = x2 + 2xy dy
. (b) 2 = x2 – 2xydy
(C) x2 = y2 + xy dy
(d) x2 = y2 + 3xy and
a
12
490 If a curve ( y=f(x) ) passes through the
point (1,-1) and satisfies the differential equation, ( boldsymbol{y}(mathbf{1}+boldsymbol{x} boldsymbol{y}) boldsymbol{d} boldsymbol{x}= )
( x d y, ) then ( fleft(-frac{1}{2}right) ) is equal to
12
491 The order and degree of the differential
equation ( y^{prime}+left(y^{prime prime}right)^{2}=left(x+y^{prime prime}right)^{2} ) are:
A .2,1
в. 1,1
( c cdot 2,2 )
D. 1,2
12
492 The D.E whose solution is ( y=a x^{3}+b x^{2} )
is
A ( cdot x^{2} y_{2}-4 x y_{1}+6 y=0 )
B . ( x^{2} y_{2}-4 y_{1}+6 y=0 )
c. ( x^{2} y_{2}+4 x y_{1}-6 y=0 )
D. ( x^{2} y_{2}-2 x y_{1}+6 y=0 )
12
493 If ( boldsymbol{y}=boldsymbol{a} e^{4 x}+boldsymbol{b} e^{5 x}, ) then prove ( boldsymbol{y}_{2}- )
( mathbf{9} boldsymbol{y}_{1}+mathbf{2 0} boldsymbol{y}=mathbf{0} )
12
494 The degree of the differential equation
( boldsymbol{x}=mathbf{1}+frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+frac{mathbf{1}}{mathbf{2 !}}left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}+ )
( frac{1}{3 !}left(frac{d y}{d x}right)^{3}+dots dots ) is?
( A cdot 3 )
B.
c. Not defined
D. None of these
12
495 Solve:
( (x-y)^{2} frac{d y}{d x}=a^{2} )
12
496 If ( (2+sin x) frac{d y}{d x}+(y+1) cos x=0 ) and
( boldsymbol{y}(mathbf{0})=mathbf{1}, ) then ( boldsymbol{y}left(frac{boldsymbol{pi}}{mathbf{2}}right) ) is equal to:
A ( cdot frac{1}{3} )
в. ( -frac{2}{3} )
( c cdot-frac{1}{3} )
D.
12
497 Solve the differential equation ( left(1-x^{2}right) frac{d y}{d x}+x y=a )
A ( .-y=-a x+c sqrt{left(1-x^{2}right)} )
В . ( y=-a x+c sqrt{left(1-x^{2}right)} )
C. ( -y=a x+c sqrt{left(1-x^{2}right)} )
D. None of these
12
498 Family ( y=A x+A^{4} ) of curves is
represented by the differential equation of degree.
( A cdot 3 )
B. 2
( c cdot 4 )
D.
12
499 The population ( p(t) ) at time ( t ) of a certain mouse species satisfies the differential equation ( frac{boldsymbol{d} boldsymbol{p}(boldsymbol{t})}{boldsymbol{d} boldsymbol{t}}= )
( 0.5 p(t)-450 . ) If ( p(0)=850, ) then the time at which the population becomes
zero is
A ( cdot frac{1}{2} ln 18 )
B. ( ln 18 )
c. ( 2 ln 18 )
D. ( ln 9 )
12
500 Solve the following differential equation
( left(cot ^{-1} y+xright) d y=left(1+y^{2}right) d x )
12
501 I.F. of ( frac{d y}{d x}+y tan x=x^{2} cos ^{2} x ) is:
A . sec
B. ( cos x )
( mathbf{c} cdot sec ^{2} x )
D. ( cos ^{2} x )
12
502 Form the differential equation corresponding to ( y^{2}-2 a y+x^{2}=a^{2} b y )
eliminating a.
12
503 The radius of water cone at ( t=1 ) is
( mathbf{A} cdot R[1-k / H] )
B . ( R[1-H / k] )
( mathbf{c} cdot R[1+H / k] )
D cdot ( R[1=k / H] )
12
504 The differential equation by eliminating ( A, B, C ) from ( y=A e^{2 x}+B e^{3 x}+C e^{2 x} )
is
12
505 If y=y(x) and
2+ sin x
y+1
= -cos x, y(0) = 1,
then y() equals
(2004)
(d) 1
(a) 1/3
(b) 2/3
(c) -1/3
12
506 The value of ( sum_{i=1}^{10} r(i) ) is equal to
A ( cdot log left[2-frac{k}{H}right] )
в. ( 5 Rleft[2+frac{11 k}{H}right] )
c. ( _{5 R}left[2-frac{11 k}{H}right] )
D. ( 4 Rleft[2-frac{11 k}{H}right. )
12
507 A conic ( C ) satisfies the differential
equation ( left(1+y^{2}right) d x-x y d y=0 ) and
passes through the point (1,0) An ellipse ( boldsymbol{E} ) which is confocal with ( boldsymbol{C} ) having its eccentricity equal to ( sqrt{2 / 3} )
(a) Find the length of the latus rectum
of the conic ( C )
(b) Find the equation of the ellipse ( boldsymbol{E} )
(c) Find the locus of the point of intersection of the perpendicular
tangents to the ellipse ( boldsymbol{E} )
12
508 f ( y=left(tan ^{-1} xright)^{2}, ) show that ( (1+ )
( left.x^{2}right)^{2} frac{d^{2} y}{d x^{2}}+2 xleft(1+x^{2}right) frac{d y}{d x}-2=0 )
12
509 Solve ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}+boldsymbol{y} ) 12
510 Write the degree of the differential
equation ( left(frac{d y}{d x}right)^{4}+3 x frac{d^{2} y}{d x^{2}}=0 )
12
511 The current in an electrical circuit is
given by ( boldsymbol{R} boldsymbol{i}+frac{boldsymbol{q}}{boldsymbol{c}}=frac{boldsymbol{E}}{boldsymbol{R}} . ) The expression
for ( boldsymbol{q} ) is?
A ( cdot q=E Cleft(1-e^{t / C R}right) )
в. ( q=E Cleft(1+e^{-t / C R}right) )
c. ( q=E Cleft(1-e^{-C R / t}right) )
D ( cdot q=E Cleft(1-e^{-t / C R}right) )
12
512 Show that ( y=frac{1}{x} ) is a solution of the differential equation ( frac{d y}{d x}=log x ) 12
513 Let ( boldsymbol{f}left(boldsymbol{x}, boldsymbol{y}, boldsymbol{c}_{1}right)=mathbf{0} ) and ( boldsymbol{f}left(boldsymbol{x}, boldsymbol{y}, boldsymbol{c}_{2}right)=mathbf{0} )
define two integral curves of a homogeneous first order differential
equation. If ( P_{1} ) and ( P_{2} ) are respectively the points of intersection of these
curves with an arbitrary line, ( boldsymbol{y}=boldsymbol{m} boldsymbol{x} )
then prove that the slopes of these two
curves at ( P_{1} ) and ( P_{2} ) are equal
12
514 The general solution of the differential equation ( frac{boldsymbol{y} boldsymbol{d} boldsymbol{x}-boldsymbol{x} boldsymbol{d} boldsymbol{y}}{boldsymbol{y}}=mathbf{0} ) is
A ( . x y=C )
В. ( x=C y^{2} )
c. ( y=C x )
D. ( y=C x^{2} )
12
515 Assertion
The order of the differential equation
whose solution is ( boldsymbol{y}=boldsymbol{c}_{1} boldsymbol{e}^{2 boldsymbol{x}+boldsymbol{c}_{2}}+ )
( c_{3} e^{2 x+c_{4}} ) is 4
Reason
Order of the differential equation is the order of the highest order derivative occurring in the differential equation.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
516 Particular solution of differential
equation ( e^{frac{d y}{d x}}=x ; y(1)=3 ; x>0 ) is
A ( cdot log y=x^{2}+4 )
B. ( y=ln x-x+4 )
c. ( y^{2}=log x+4 )
D. ( 2 y=x^{2}+5 )
E. ( y=x ln x-x+4 )
12
517 Obtain a differential equation by
eliminating ( c ) when it is given
( tan ^{-1} x+tan ^{-1} y=tan ^{-1} c )
12
518 Solve the following differential eqauton
( boldsymbol{y}^{2} boldsymbol{d} boldsymbol{y}-boldsymbol{x}^{2} boldsymbol{d} boldsymbol{x}=mathbf{0} )
12
519 ff ( y sqrt{1-x^{2}}+x sqrt{1-y^{2}}=1 ) then ( frac{d y}{d x}= )
A ( cdot-sqrt{frac{1-y^{2}}{1-x^{2}}} )
B. ( sqrt{frac{1-y^{2}}{1-x^{2}}} )
c. ( sqrt{frac{1-x^{2}}{1-y^{2}}} )
D. None of these
12
520 23.
If y = y(x) is the solution of the differential equation
y + 2y = x’ satisfying y(a)= 1, then y is equal to:
[JEE M 2019-9 Jan (M)
(b)
12
521 The solution of ( frac{d y}{d x}=|x| ) is :
A ( cdot y=frac{x|x|}{2}+c )
В . ( y=frac{|x|}{2}+c )
C・ ( y=frac{x^{2}}{2}+c )
D・ ( y=frac{x^{3}}{2}+c )
12
522 Reduce each of the following differentia equations to the variables separable
( i ) ) ( 1+frac{d y}{d x}= )
form and hence solve.
( operatorname{cosec}(boldsymbol{x}+boldsymbol{y}) )
ii) ( (x-y)^{2} frac{d y}{d x}=a^{2} )
12
523 The solution of ( frac{d y}{d x}=e^{(y-x)} ) is
A ( cdot e^{y}+e^{x}=c )
B . ( e^{-x}=e^{-y}+c )
c. ( e^{y-x}=c )
D ( cdot e^{y / x}=c )
12
524 What is the solution of the differential
equation ( boldsymbol{x} boldsymbol{d} boldsymbol{y}+boldsymbol{y} boldsymbol{d} boldsymbol{x}=mathbf{0} )
A ( . x y=c )
в. ( y=c x )
c. ( x+y=c )
D. ( x-y=c )
12
525 The solution of ( (x+y+1) frac{d y}{d x}=1 ) is:
A ( . x=-(y+2)+c e^{y} )
B . ( y=-(x+2)+c e^{x} )
c. ( x=-(y+2)+c e^{x} )
D. ( x=(y+2)+c e^{-y} )
12
526 A population grows at the rate of ( 8 % ) per year. How long does it take for the
population to double ? Use differential equation for it.
12
527 D.E. of family of parabolas symmetrical
about the X-axis is
A ( cdot y frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}=0 )
B. ( y frac{d^{2} y}{d x^{2}}-left(frac{d y}{d x}right)^{2}=0 )
( ^{mathbf{C}}-y frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}=1 )
D. ( y^{2} frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}=1 )
12
528 Solution of the differential equation
( tan y cdot sec ^{2} x d x+tan x cdot sec ^{2} y d y=0 ) is
A. ( tan x+tan y=k )
B. ( tan x-tan y=k )
c. ( frac{tan x}{tan y}=k )
D. ( tan x cdot tan y=k )
12
529 Form the differential equation from the following primitives where constant is arbitrary.
( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c} )
12
530 Solve: ( boldsymbol{y}^{prime}=frac{mathbf{3} boldsymbol{x}-boldsymbol{y}}{boldsymbol{x}+boldsymbol{y}} )
( mathbf{A} cdot(3 x+y)(x-y)=c_{0} )
B ( cdot(3 x-y)(x+y)=c_{0} )
( mathbf{c} cdot(x-3 y)(3 x+y)=c_{0} )
( mathbf{D} cdot(x+3 y)(x-y)=c_{0} )
12
531 The differential equation for which ( sin ^{-1} x+sin ^{-1} y=c ) is given by:
A ( cdot sqrt{1-x^{2}} d y+sqrt{1-y^{2}} d x=0 )
B ( cdot sqrt{1-x^{2}} d x+sqrt{1-y^{2}} d y=0 )
C ( . sqrt{1-x^{2}} d x-sqrt{1-y^{2}} d y=0 )
D. ( sqrt{1-x^{2}} d y-sqrt{1-y^{2}} d x=0 )
12
532 Check whether the function is
homogenous or not. If yes then find the degree of the function
( g(x)=x^{2}-8 x^{3} )
A. Not homogenous
B. Homogenous with degree 4
c. Homogenous with degree 2
D. None of these
12
533 A normal is drawn at a point P(x, y) of a curve. It meets the
x-axis at Q. If PQ is of constant length k, then show that the
differential equation describing such curves is
(1994 – 5 Marks)
Find the equation of such a curve passing through (0, k).
12
534 A tank contains 100 liters of fresh water.
A solution containing 1 gm/litre of soluble lawn fertilizer runs into the tank at the rate of 1 lit/min., and the mixture
is pumped out of the tank at the rate of
3 litres/min. Find the time when the
amount of fertilizer in the tank is
maximum.

Enter ( 9 t ) i.e, ( (t ) is time in minutes)

12
535 A differential equation representing the
family of curves ( boldsymbol{y}=boldsymbol{a} sin (boldsymbol{lambda} boldsymbol{x}+boldsymbol{alpha}) ) is:
( ^{A} cdot frac{d^{2} y}{d x^{2}}+lambda^{2} y=0 )
B. ( frac{d^{2} y}{d x^{2}}-lambda^{2} y=0 )
c. ( frac{d^{2} y}{d x^{2}}+lambda y=0 )
D. None of the above
12
536 If ( frac{sin ^{-1} x}{sqrt{1-x^{2}}}=y, ) show that ( (1- )
( left.x^{2}right) frac{d^{2} y}{d x^{2}}-3 x frac{d y}{d x}-y=0 )
12
537 State whether the following statement is True or False?

The differential equation of a circuit
with inductance ( L ) and resistance ( R ) is
( operatorname{givenby} frac{boldsymbol{d} boldsymbol{i}}{boldsymbol{d} boldsymbol{t}}+frac{boldsymbol{R}}{boldsymbol{L}} boldsymbol{i}=frac{boldsymbol{E}}{boldsymbol{L}} boldsymbol{e}^{-boldsymbol{a t}}(boldsymbol{i}= )
0 at ( t=0 ). The current at time ( t ) is given
by ( i=frac{E}{R-a L}left(e^{-a t}right) )
A . True
B. False

12
538 The differential equation of all nonhorizontal lines in a plane is:
A ( cdot frac{d^{2} y}{d x^{2}}=0 )
B. ( frac{d^{2} x}{d y^{2}}=0 )
c. ( frac{d y}{d x}=0 )
D. ( frac{d x}{d y}=0 )
12
539 If ( boldsymbol{y}=sqrt{(boldsymbol{a}-boldsymbol{x})(boldsymbol{x}-boldsymbol{b})}-(boldsymbol{a}- )
( b) tan ^{-1} sqrt{frac{a-x}{x-b}}(a>b) ) then ( frac{d y}{d x}= )
A. ( sqrt{frac{a-x}{x-b}} )
B . ( sqrt{(a-x)(x-b)} )
c. 0
D.
12
540 The solution of ( y frac{d y}{d x}=1+y^{2} ) is:
A ( cdot 2 x=log left[cleft(1+y^{2}right)right] )
в. ( x=c y^{2} )
c. ( cleft(1+y^{2}right)=x )
D ( cdot 2 y=log left{cleft(1+x^{2}right)right} )
12
541 I.F. of ( frac{d y}{d x}=y tan x+2 sin x ) is:
A . sec ( x )
B. ( sin x )
( c cdot csc x )
D. ( cos x )
12
542 f ( y=sin left(2 sin ^{-1} xright), ) then prove that
( frac{d y}{d x}=2 sqrt{frac{1-y^{2}}{1-x^{2}}} )
12
543 Solve the following differential equation
( frac{d y}{d x}+1=sin x )
12
544 Find an expression for ( boldsymbol{y} ) given ( frac{d y}{d x}=7 x^{5} ) 12
545 Prove that ( y=a e^{-2 x}+b e^{x} ) is the
solution of differential equation ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+ )
( frac{d y}{d x}-2 y=0 )
12
546 Solve the following differential equation ( frac{d y}{d x}=1-cos x ) 12
547 For the following differential equation,
find the general solution.
( boldsymbol{x} boldsymbol{d} boldsymbol{y}=boldsymbol{d} boldsymbol{x} )
12
548 ( left(x^{3}-y^{3}right) d x+x y^{2} d y=0 . ) Solving this
we get ( frac{boldsymbol{k}}{boldsymbol{x}}=boldsymbol{e}^{boldsymbol{y}^{m} / boldsymbol{n} boldsymbol{x}^{r}} . ) Find ( boldsymbol{m}+boldsymbol{n}+boldsymbol{r} ? )
12
549 Solve the differential equation ( y d x+ )
( left(boldsymbol{x}-boldsymbol{y} boldsymbol{e}^{boldsymbol{y}}right) boldsymbol{d} boldsymbol{y}=mathbf{0} )
12
550 Order of differential equation of ( y= )
( boldsymbol{m} boldsymbol{x}+boldsymbol{c} )
12
551 Radium disappears at a rate
proportional to the amount present. If ( 5 % ) of the original amount disappears
in 50 years, how much will remain at
the end of 100 years. ( left[text { Take } A_{0} ) as the right. initial amount].
12
552 Solve: ( 2 x y d x+left(x^{2}+2 y^{2}right) d y=0 ) 12
553 ( y+x^{2}=frac{d y}{d x} ) has the solution
A. ( y+x^{2}+2 x+2=c . e^{x} )
B . ( y+x+2 x^{2}+2=c . e^{x} )
C ( cdot y+x+x^{2}+2=c cdot e^{2 x} )
D . ( y^{2}+x+x^{2}+2=c . e^{2 x} )
12
554 Form the differential equation
representing the family of curves ( y^{2}= )
( mleft(a^{2}-x^{2}right), ) where ( a ) and ( m ) are
parameters.
12
555 Find the degree of each algebraic
expression ( boldsymbol{p} boldsymbol{q}+boldsymbol{p}^{2} boldsymbol{q}-boldsymbol{p}^{2} boldsymbol{q}^{2} )
12
556 11. Let I denote a curve y=y(x) which is in the first quadrant
and let the point (1, 0) lie on it. Let the tangent to at a point
Pintersect the y-axis at Y. If PY, has length 1 for each point
Pon T, then which of the following options is/are correct?
(JEE Adv. 2019
(2) xy’t V1 – x² = 0
12
557 Solve the given differential equation
( left(x y^{2}+xright) d x+left(y x^{2}+yright) d y=0 )
( mathbf{A} cdotleft(x^{2}+1right)left(y^{2}+1right)=c )
B ( cdot log left(x^{2}+1right) log left(y^{2}+1right)=c )
c. ( left(x^{2}+1right)+left(y^{2}+1right)=c )
D. none of these
12
558 The population of a city increases at the rate of ( 4 % ) per year. If in time ( t, ) the
population becomes ( p ), then equation of
( p ) in terms of ( t ) is
A.
[
p=e^{frac{t}{25}}
]
В. ( quad p=4 . e^{frac{t}{25}} )
c. ( quad_{p=c . e} frac{t}{25} )
D.
[
p=frac{1}{25} e^{4 t}
]
12
559 Find the particular solution of the differential equation ( log left(frac{d y}{d x}right)=3 x+ )
4 ( y ), given that ( y=0 ) when ( x=0 )
12
560 I.F of ( x frac{d y}{d x}=left(2 y+2 x^{4}+x^{2}right) ) is:
A . ( x^{-2} )
B . ( x^{-1} )
c. ( x )
D. ( x^{2} )
12
561 The differential equation of the family of circles passing through the fixed points ( (a, 0) ) and ( (-a, 0) ) is:
A ( cdot y_{1}left(y^{2}-x^{2}right)+2 x y+a^{2}=0 )
B . ( y_{1} y^{2}+x y+a^{2} x^{2}=0 )
C. ( y_{1}left(y^{2}-x^{2}+a^{2}right)+2 x y=0 )
D. none of these
12
562 A body at an unknown temperature is placed in a room which is held at a
constant temperature of ( 30^{0} F . ) If after
10 minutes the temperature of the body
is ( 0^{0} F ) and after 20 minutes the
temperature of the body is ( 15^{0} F, ) find the unknown initial temperature.
A . -50
B. -20
c. -40
D. -30
12
563 The bacteria culture grows at a rate
proportional to its size. After 2 hours
there are 600 bacteria and after 8
hours the count is 75000 . Find the
initial population and when the
population reach ( 200000 ? )
12
564 Show that the differential equation ( (x-y) frac{d y}{d x}=x+2 y ) is homogeneous 12
565 The sum of Rs. 1000 is compounded continuously, the nominal rate of interest being four percent per annum. In how many years will the amount be
twice the original principal? ( left(log _{e} 2=right. )
( mathbf{0 . 6 9 3 1}) )
12
566 If ( boldsymbol{y}(boldsymbol{x}) ) satisfies the differential equation ( cos x frac{d y}{d x}-y sin x=6 x ) and
( boldsymbol{y}left(frac{boldsymbol{pi}}{mathbf{3}}right)=mathbf{0 .} ) Then value of ( boldsymbol{y}left(frac{boldsymbol{pi}}{boldsymbol{6}}right) ) is
A ( cdot frac{pi^{2}}{3 sqrt{2}} )
в. ( frac{-pi^{2}}{3 sqrt{2}} )
c. ( frac{pi^{2}}{2 sqrt{3}} )
D. ( frac{pi^{2}}{4} )
12
567 Solve ( e^{x} tan y d x+left(1-e^{x}right) sec ^{2} y d y= )
0
12
568 What is the general solution of the
differential equation ( e^{x} tan y d x+ )
( left(1-e^{x}right) sec ^{2} y d y=0 ? )
A・sin ( y=cleft(1-e^{x}right) ) where ( c ) is the constant of integration
B cdot ( cos y=cleft(1-e^{x}right) ) where ( c ) is the constant of integration
C ( cdot cot y=cleft(1-e^{x}right) ) where ( c ) is the constant of integration
D. None of the above
12
569 If ( boldsymbol{x}=tan left(e^{-y}right), ) then show that ( frac{d y}{d x}= )
( frac{e^{-y}}{1+x^{2}} )
12
570 Form the differential equation by eliminating arbitrary constants from the relation ( boldsymbol{y}=boldsymbol{A} boldsymbol{e}^{boldsymbol{5} boldsymbol{x}}+boldsymbol{B} boldsymbol{e}^{-boldsymbol{5} boldsymbol{x}} ) 12
571 The order of the differential equation
whose solution is
( boldsymbol{y}=boldsymbol{c}_{1} boldsymbol{e}^{boldsymbol{x}}+boldsymbol{c}_{2} sin boldsymbol{x}+boldsymbol{c}_{3} cos boldsymbol{x} ) is
( A cdot 4 )
B. 3
( c cdot 2 )
( D )
12
572 Solve:
( x^{3} frac{d y}{d x}=y^{3}+y^{2} sqrt{y^{2}-x^{2}} )
A ( . x y=c(x-sqrt{y^{2}-x^{2}}) )
B . ( x=cleft(y+y^{2}-x^{2}right) )
c. ( y=cleft(x+y^{2}-x^{2}right) )
D. ( x y=c(y+sqrt{y^{2}-x^{2}}) )
12
573 The solution of the differential equation ( x d y-y d x=sqrt{x^{2}-y^{2}} d x ) is 12
574 Solution of the differential equation ( frac{d y}{d x}+y sec x=tan xleft(leq x<frac{pi}{2}right) ) is
A ( cdot y(sec x-tan x)=(sec x+tan x)-x+C )
B cdot ( y(sec x+tan x)=(sec x-tan x)-x+C )
C ( cdot y(sec x+tan x)=(sec x+tan x)-x+C )
D. noneofthese
12
575 2.
The solution of the equation a y
12
– + cxtd
cos
2* + cx? + d.
+ cx
(a) ** + cx+d
+ cx + d
12
576 If ( boldsymbol{y}=boldsymbol{a}+boldsymbol{b} boldsymbol{x}^{2} ; mathbf{a}, mathbf{b} ) are arbitrary
constants then:
A ( cdot frac{d^{2} y}{d x^{2}}=2 x y )
B. ( x frac{d^{2} y}{d x^{2}}=frac{d y}{d x} )
c. ( x frac{d^{2} y}{d x^{2}}-frac{d y}{d x}+y=0 )
D. ( frac{d^{2} y}{d x^{2}}=2 y )
12
577 Find ( frac{d y}{d x} ) when ( y=e^{sqrt{x}+sin x} ) 12
578 Which of the following is/are correct for the function ( x cos frac{y}{x} ? )
This question has multiple correct options
A. Not homogenous
B. Homogenous
c. Degree ( =1 )
D. Degree=2
12
579 Degree of ( left[2+left(frac{d y}{d x}right)^{2}right]^{3 / 2}=a frac{d^{2} y}{d x^{2}} ) is:
( A )
B.
( c )
( D )
12
580 Find the differential equation of the family of all straight lines passing through the origin. 12
581 Find the order and degree, if defined of
the differential equation
( left(frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}right)^{3}+left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}+sin frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+mathbf{1}=mathbf{0} )
12
582 Solve ( (boldsymbol{x} boldsymbol{y}-mathbf{1}) frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{y}^{2}=mathbf{0} )
A. ( x y+log x=C )
в. ( x y+log y=C )
c. ( x y-log y=C )
D. ( x y-log x=C )
12
583 The charge ( q ) on a plate of a condenser
of capacity ( C ) charged through a
resistance ( boldsymbol{R} ) by a steady voltage ( boldsymbol{V} ) satisfies the differential equation ( R frac{d q}{d t}+frac{q}{c}=V . ) If ( q=0 ) at ( t=0 ) then the
expression for the charge on the condenser is?
A ( cdot q=C Vleft(1-cos frac{-t}{R C}right) )
В cdot ( q=C Vleft(1-sin frac{-t}{R C}right) )
c. ( q=C Vleft(1-e^{-t / R C}right. )
D・ ( q=C Vleft(1-cot frac{-t}{R C}right) )
12
584 Form the differential equation corresponding to ( y^{2}-2 a y+x^{2}=a^{2} b y )
eliminating a.
12
585 Assertion
The order of the differential equation whose primitive is ( y=A+ln B x ) is 2
Reason
f there are ‘n’ independent arbitrary constant in a family of curve then the order of the corresponding differential equation is ‘n’
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
586 The population of a town grows at the
rate of ( 10 % ) per year. Using differential equation, find how long will it take for the population to grow 4 times.
12
587 Solve the differential equation ( left(x^{2}+right. )
( left.boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{x}-boldsymbol{2} boldsymbol{x} boldsymbol{y} boldsymbol{d} boldsymbol{y}=mathbf{0} )
12
588 Find the general solution of the differential equation ( frac{d y}{d x}+y . c o t x= )
( 2 x+x^{2} cdot cot x )
12
589 The solution of the differential equation
(1+y)+ (x – e tandy – a
[2003]
(a) re2tanly = etan ‘y+k (b) (x – 2) = ke2 tany
(c) 2xetany = e2 tany +k (d) xetan’ y = tany+k
12
590 Write the order and degree of the differential equation ( boldsymbol{y}=boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+ )
( boldsymbol{a} sqrt{1+left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}} )
12
591 U
UN
11.
The soluton of the differential equation
dy x+y
dy = satisfying the condition y(1)=1 is
(a) y= ln x+x
(b) y=x In x + x2
(c) y = xex – 1).
(d) y=x In x + x
12
592 Show that the differential equation ( x frac{d y}{d x} sin frac{y}{x}+x-y sin frac{y}{x}=0 ) is
homogeneous. Find the particular solution of this differential equation given that ( x=1 ) when ( y=frac{pi}{2} )
12
593 The order of the differentiable equation
associated with the primitive ( boldsymbol{y}= )
( C_{1}+C_{2} e^{x}+C_{3} e^{-2 x+C_{4}} ) where
( C_{1}, C_{2}, C_{3}, C_{4} ) are arbitrary constants,
is
A . 3
B. 4
( c cdot 2 )
D. None of these
12
594 The differential equation of all conics
whose axes coincide with the co-
ordinate axes, is
A ( . x y y_{2}+x y_{1}^{2}-y y_{1}=0 )
B . ( y y_{2}+y_{1}^{2}-y y_{1}=0 )
c. ( x y y_{2}+(x-y) y_{1}=0 )
D. ( left(y y_{1}right)^{2}-x y_{1}-y x=0 )
12
595 A bacteria population increases sixfold
in 10 hours. Assuming normal growth, how long did it take for their population
to double?
A. 3.93 hrs
B. 3.87 hrs
( c .3 .72 ) hrs
D. 3.54 hrs
12
596 A body is propelled straight up with an initial velocity of ( 500 mathrm{ft} / mathrm{sec} ) in a vacuum with no air resistance. How
long will it take the body to return to the ground (in seconds)?
( mathbf{A} cdot 21.25 )
B. 31.25
c . 35.25
D. 25.25
12
597 Form the different equations of all concentric circles at the origin. 12
598 Find the general solution of the differential equation ( frac{d y}{d x}-y=cos x ) 12
599 Find the order and degree of the
differential equations:
( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}=left[boldsymbol{y}+left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{6}right]^{1 / 4} )
12
600 The differential equation of the family of curves ( boldsymbol{y}=boldsymbol{A} boldsymbol{e}^{boldsymbol{3} boldsymbol{x}}+boldsymbol{B} boldsymbol{e}^{boldsymbol{5} boldsymbol{x}}, ) where ( boldsymbol{A} ) and
( B ) are arbitrary constants, is
A ( cdot frac{d^{2} y}{d x^{2}}+8 frac{d y}{d x}+15 y=0 )
B. ( frac{d^{2} y}{d x^{2}}-frac{d y}{d x}+y=0 )
c. ( frac{d^{2} y}{d x^{2}}-8 frac{d y}{d x}+15 y=0 )
D. None of the above
12
601 The integrating factor of the differential equation ( boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{y}=boldsymbol{x}^{3} boldsymbol{y}^{boldsymbol{6}} ) is
A ( cdot-frac{5}{x^{5}} )
в. ( frac{5}{y^{5}} )
( c cdot-frac{1}{x^{5}} )
D. ( frac{1}{x^{5}} )
12
602 f ( m ) and ( n ) are the order and degree of
the differential equation ( left(frac{d^{2} y}{d x^{2}}right)^{5}+4 frac{left(frac{d^{2} y}{d x^{2}}right)^{3}}{left(frac{d^{2} y}{d x^{2}}right)}+frac{d^{2} y}{d x^{2}}=x^{2}-1 )
then
A ( . m=3, n=3 )
B. ( m=2, n=6 )
( mathrm{c} cdot m=3, n=5 )
12
603 By eliminating the arbitrary constants A and ( mathrm{B} ) from ( boldsymbol{y}=boldsymbol{A} boldsymbol{x}^{2}+boldsymbol{B} boldsymbol{x}, ) we get the
differential equation:
A ( cdot frac{d^{3} y}{d x^{3}}=0 )
B. ( x^{2} frac{d^{2} y}{d x^{2}}-2 x frac{d y}{d x}+2 y=0 )
c. ( frac{d^{2} y}{d x^{2}}=0 )
D. ( x^{2} frac{d^{2} y}{d x^{2}}+y=0 )
12
604 Form the differential equation from the following primitive where constant is arbitrary. ( boldsymbol{x} boldsymbol{y}=boldsymbol{a}^{2} ) 12
605 The differential equation of the family of curves ( v=frac{A}{r}+B, ) where ( A ) and ( B ) are
arbitrary constants is
A ( cdot frac{d^{2} v}{d r^{2}}+frac{1}{r} frac{d v}{d r}=0 )
B. ( frac{d^{2} v}{d r^{2}}-frac{2}{v} frac{d v}{d r}=0 )
c. ( frac{d^{2} v}{d r^{2}}+frac{2 d v}{r d r}=0 )
D. ( frac{d^{2} v}{d r^{2}}+frac{2}{v} frac{d v}{d r}=0 )
12
606 1.
The order of the differential equation whose general solution
is given by
y=(C+C) cos(r=C2)-Cercs, where C.C.C.ca
Ce are arbitrary constants, is
(1998-2 Marks)
12
607 Form the differential equation of the
family of curves represented by ( y^{2}= )
( (x-c)^{3} )
12
608 If the differential equation representing the family of all circles touching ( x- ) axis at the origin is ( left(x^{2}-y^{2}right) frac{d y}{d x}= )
( boldsymbol{g}(boldsymbol{x}) boldsymbol{y}, ) then ( boldsymbol{g}(boldsymbol{x}) ) equals:
A ( cdot frac{1}{2} x )
в. ( 2 x^{2} )
c. ( 2 x )
D. ( frac{1}{2} x^{2} )
12
609 Solve:
( left(x frac{d y}{d x}-yright) tan ^{-1} frac{y}{x}=x )
12
610 Differential equation from ( a x^{2}+b y^{2}= )
( mathbf{1} ) is
12
611 Find the degree of the differential equation: ( sqrt{1+left(frac{d y}{d x}right)^{2}}=4 x )
( A cdot 1 )
B. 2
( c cdot 4 )
D. None of these
12
612 Solve the following differential equation ( cos x frac{d y}{d x}-cos 2 x=cos 3 x ) 12
613 Form the differential equation
corresponding to ( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x},(boldsymbol{a}, boldsymbol{b}) )
12
614 If ( cos y=x cos (a+y) ) then prove that:
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{cos ^{2}(boldsymbol{a}+boldsymbol{y})}{sin boldsymbol{a}} )
12
615 The population ( p(t) ) a time ( t ) of a certain mouse species satisfies the differential equation ( frac{boldsymbol{d} boldsymbol{p}(boldsymbol{t})}{boldsymbol{d} boldsymbol{t}}=frac{1}{2} boldsymbol{p}(boldsymbol{t})-boldsymbol{4 5 0} )
( boldsymbol{p}(boldsymbol{0})=mathbf{8 5 0}, ) then the time at which the
population becomes zero is:
A ( .2 ~ ln 18 )
B. ( ln 9 )
c. ( frac{1}{2} ln 18 )
D. ( ln 18 )
12
616 If ( boldsymbol{y}=cos ^{-1} boldsymbol{x} ) then ( left(1-boldsymbol{x}^{2}right) boldsymbol{y}^{prime prime}-boldsymbol{x} boldsymbol{y}^{prime}=? )
where ( boldsymbol{y}^{prime}=frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
A . 1
B.
( c cdot 2 x )
D. ( 2 y )
12
617 The population ( P ) of a town decreases at a rate proportional to the number by which the population exceeds-10 30, proportionality constant being ( mathrm{k}>0 ) Find Population at any time ( t ) given initial population of the town being ( 2500 ., ) then ( P=1000+1500 e^{-k t} )
then ( k=? )
A ( cdot k=frac{1}{10} ln left(frac{5}{3}right) )
B . ( k=frac{-1}{10} ln left(frac{5}{3}right) )
c. ( k=ln left(frac{5}{3}right) )
D cdot ( k=frac{1}{10} ln left(frac{3}{5}right) )
12
618 The solution of ( frac{d y}{d x}=x log x )
A ( cdot 2 y=x^{2}left[log x+frac{1}{2}right]+c )
B. ( 2 y=x^{2}left[log x-frac{1}{2}right]+c )
c. ( _{y}=frac{x^{2}}{2}(log 2-x)+c )
D. ( y^{2}=x^{2} log x+x+c )
12
619 A solution of the differential equation ( left(frac{d y}{d x}right)^{2}-x frac{d y}{d x}+y=0 )
( mathbf{A} cdot y=2 )
в. ( y=2 x )
( mathbf{c} cdot y=2 x-4 )
D. ( y=2 x^{2}-4 )
12
620 Write the order of the differential
equation associated with the primitive ( boldsymbol{y}=boldsymbol{C}_{1}+boldsymbol{C}_{2} boldsymbol{e}^{boldsymbol{x}}+boldsymbol{C}_{3} boldsymbol{e}^{-boldsymbol{2} boldsymbol{x}+boldsymbol{C}_{4}}, ) where
( C_{1}, C_{2}, C_{3}, C_{4} ) are arbitrary constants.
12
621 Obtain the differential equation by
eliminating the arbitrary constants from the following equation:
( boldsymbol{y}=boldsymbol{c}_{1} boldsymbol{e}^{2 boldsymbol{x}}+boldsymbol{c}_{2} boldsymbol{e}^{-boldsymbol{2} boldsymbol{x}} )
12
622 The degree of the differential equation ( left(1+left(frac{d y}{d x}right)^{2}right)^{3 / 4}=left(frac{d^{2} y}{d x^{2}}right)^{1 / 3} ) is:
( A )
B.
( c cdot 9 )
( D )
12
623 The general solution of the differentia equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}+boldsymbol{y}+mathbf{1}}{mathbf{2} boldsymbol{x}+mathbf{2} boldsymbol{y}+mathbf{1}} ) is:
A ( cdot log _{e}|3 x+3 y+2|+3 x+6 y=C )
B . ( log _{e}|3 x+3 y+2|-3 x+6 y=C )
( mathbf{c} cdot log _{e}|3 x+3 y+2|-3 x-6 y=C )
D ( cdot log _{e}|3 x+3 y+2|+3 x-6 y=C )
12
624 If ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+frac{boldsymbol{3}}{cos ^{2} boldsymbol{x}}=frac{1}{cos ^{2} boldsymbol{x}}, boldsymbol{x} boldsymbol{epsilon}left(frac{-boldsymbol{pi}}{boldsymbol{3}}, frac{boldsymbol{pi}}{boldsymbol{3}}right) )
and ( yleft(frac{pi}{4}right)=frac{4}{3}, ) then ( yleft(-frac{pi}{4}right) ) equals:
A ( cdot frac{16}{3} )
B. ( frac{1}{3} )
c. ( frac{-4}{3} )
D. ( frac{1}{3}+e^{3} )
12
625 If ( cos x ) and ( sin x ) are solutions of the
different equation ( boldsymbol{a}_{mathbf{0}} frac{boldsymbol{d}^{mathbf{2}} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+boldsymbol{a}_{1} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+ )
( boldsymbol{a}_{2} boldsymbol{y}=mathbf{0}, ) where ( boldsymbol{a}_{0}, boldsymbol{a}_{1}, boldsymbol{a}_{2} ) are real
constants, then which of the following
is/are always true?
This question has multiple correct options
( mathbf{A} cdot A cos x+B sin x ) is a solution, where ( A ) and ( B ) are real
costants
B. ( A cos left(x+frac{pi}{4}right) ) is a solution, where ( A ) is real constant
c. ( A cos x sin x ) is a solution, where ( A ) is real constant
D. ( A cos left(x+frac{pi}{4}right)+B sin left(x-frac{pi}{4}right) ) is a solution, where ( A )
and ( B ) are real constants
12
626 Time required for coffee to have ( 105^{circ} mathrm{F} )
temperature is
( A cdot 6 min )
B. 6.43 min
c. ( 7.23 min )
D. 7.63 min
12
627 Solve the differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( frac{x^{2}+5 x y+4 y^{2}}{x^{2}} )
12
628 The differential equation for the family of curves ( x^{2}-y^{2}+2 a x=0, ) where ( a ) is
an arbitrary constant is:
A ( cdotleft(x^{2}-y^{2}right)=2 x y y^{prime} )
B . ( x^{2}-y^{2}=-2 x y . y^{prime} )
c. ( left(x^{2}+y^{2}right) y^{prime}=2 x y )
D. ( x^{2}+y^{2}=2 x y . y^{prime} )
12
629 The differential equation of the family of
curve ( boldsymbol{y}^{2}=mathbf{4} boldsymbol{a}(boldsymbol{x}+boldsymbol{a}) ) is
A ( cdot y^{2}=4 frac{d y}{d x}left(x+frac{d y}{d x}right) )
B. ( quad y^{2}left(frac{d y}{d x}right)^{2}+2 x y frac{d y}{d x}-y^{2}=0 )
c. ( y^{2} frac{d y}{d x}+4=0 )
D. ( 2 y frac{d y}{d x}+4 a=0 )
12
630 ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{y}+boldsymbol{x} tan left(frac{boldsymbol{y}}{boldsymbol{x}}right)}{boldsymbol{x}} Rightarrow sin frac{boldsymbol{y}}{boldsymbol{x}}= )
( mathbf{A} cdot c x^{2} )
в. ( c x )
( c cdot c x^{3} )
( D cdot c x^{4} )
12
631 Solve: ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+mathbf{1}=boldsymbol{e}^{boldsymbol{x}+boldsymbol{y}} ) 12
632 The degree and order of
the differential equation
( left[1+2left(frac{d y}{d x}right)^{2}right]^{1 / 2}=5 frac{d^{2} y}{d x^{2}} ) are
( A cdot 1,2 )
B. 2,
c. 3,1
D. 4,3
12
633 The order and degree of ( left(frac{d^{2} y}{d x^{2}}right)^{1 / 3}= ) ( mathbf{1 0}+mathbf{9} boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is:
A . 2,3
B. 2,
( c cdot 1,3 )
D. 1
12
634 What is the number of arbitrary constants in the particular solution of
differential equation of third order?
A .
в.
( c cdot 2 )
D.
12
635 9.
The function y=f(x) is the solution of the differential equation
11
dx
in (-1, 1) satisfying f(0) = 0. Then
x² – 1
v
J f (x)d(x) is
(JEE Adv. 2014)
wla
12
636 Find ( frac{d y}{d x} ) if ( y=sin ^{-1}left(frac{2^{x+1}}{1+4^{x}}right) ) 12
637 Determine the order and degree(if defined) of the following differential equation. ( boldsymbol{y}^{prime prime}+left(boldsymbol{y}^{prime}right)^{2}+mathbf{2} boldsymbol{y}=mathbf{0} ) 12
638 The solution of ( frac{d^{2} x}{d y^{2}}-x=k, ) where ( k ) is a non-zero constant, vanishes when
( y=0 ) and tends of finite limit as ( y )
tends to infinity, is
( mathbf{A} cdot x=kleft(1+e^{-y}right) )
в. ( x=kleft(e^{y}+e^{-y}-2right) )
C ( . x=kleft(e^{-y}-1right) )
D. ( x=kleft(e^{y}-1right) )
12
639 Represent the following families of curves by forming the corresponding differential equation.(a, b being parameters). ( x^{2}-y^{2}=a^{2} ) 12
640 The solution of ( frac{d y}{d x}+a y=e^{m x} ) is (where ( a+m=0 ) ) is:
A ( cdot e^{a x} y=x+c )
B . ( e^{a x} y=y+c )
c. ( e^{a y} x=y+c )
D. ( e^{a y} y=x+c )
12
641 Find the differential coefficient of
( tan ^{-1} x ) w.r to ( x )
12
642 If ( frac{mathbf{d} boldsymbol{y}}{mathbf{d} boldsymbol{x}}=frac{boldsymbol{x}-boldsymbol{y}}{boldsymbol{x}+boldsymbol{y}} ) and ( boldsymbol{y}(1)=mathbf{1} ) then ( boldsymbol{y}(boldsymbol{2}) )
equals
( mathbf{A} cdot mathbf{5} )
B. ( 2 pm sqrt{10} )
c. ( 2 pm sqrt{12} )
D. ( -2 pm sqrt{10} )
12
643 ( left(frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}right)^{2}+cos left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)=mathbf{0} ) 12
644 The differential equation by eliminating
( boldsymbol{a}, boldsymbol{b} ) from ( (boldsymbol{x}-boldsymbol{a})^{2}+(boldsymbol{y}-boldsymbol{b})^{2}=boldsymbol{r}^{2} ) is
( mathbf{A} cdotleft(1-left(y_{1}right)^{2}right)^{3}=r^{2}left(y_{2}right)^{2} )
B . ( left(1+left(y_{1}right)^{2}right)^{3}=r^{2}left(y_{2}right)^{2} )
C. ( left(1+left(y_{1}right)^{2}right)^{3}=r y^{2} )
D. ( y_{1}^{2}=r y^{2} )
12
645 If ( boldsymbol{y}=e^{m x}+e^{-m x}, ) then prove that
( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}=boldsymbol{m}^{2} boldsymbol{y} )
12
646 The differential equation of all straight lines in a plane passing through (0,1) is:
A ( . y-1=m x )
в. ( y=m(x-1) )
c. ( y=x y_{1} )
D. ( y=x y_{1}+1 )
12
647 The differential equation of family of circles with fixed radius 5 units ( & )
centre lies on the line ( y=2, ) is
A ( cdot(y-2) y^{prime 2}=25-(y-2)^{2} )
B・ ( (y-2)^{2} y^{prime 2}=25-(y-2)^{2} )
c. ( (x-2) y^{prime 2}=25-(y-2)^{2} )
D・ ( (x-2) 2 y^{prime 2}=25-(y-2)^{2} )
12
648 Evaluate:
( (1+cos x) d y=(1-cos x) d x )
12
649 The degree of ( frac{d^{2} y}{d x^{2}}+ )
( left[1+left(frac{d y}{d x}right)^{2}right]^{3 / 2}=0 )
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D.
12
650 The order and degree of the differential
equation
( left[left{x-left(frac{d y}{d x}right)^{2}right}^{frac{3}{2}}right]^{2}=left(a^{2} frac{d^{2} y}{d x^{2}}right) )
( A )
B. 1,2
( c .2,2 )
D. ( 1, )
12
651 ( A & B ) are two separate reservoirs of water. The capacity of reservoir A is double the capacity of reservoir B. Both the reservoirs are filled completely with water, their inlets are closed and then
the water is released simultaneously
from both the reservoirs. The rate of flow
of water out of each reservoir at any
instant of time is proportional to the quantity of water in the reservoir at that time. One hour after the water is
released, the quantity of water in reservoir A is 1.5 times the quantity of water in reservoir B. After how many hours do both the reservoirs have the
same quantity of water? (Enter ( n ) if the answer is ( T=log _{4 / 3} n ) )
12
652 At present, a firm is manufacturing 2000 items. It is estimated
that the rate of change of production P w.r.t. additional
dP
number of workers x is given by “. = 100 – 12Vx. If the
firm employs 25 more workers, then the new level of
production of items is
[JEE M 2013]
(a) 2500 (b) 3000 () 3500 (d) 4500
12
653 If the differential equation of a body of mass ( m ) falling from rest subjected to the force of gravity and an air resistance proportional to the square of the velocity is given by ( m v frac{d v}{d x}=k a^{2}- ) ( k v^{2}, ) then it can be proved that ( frac{2 k x}{m}= ) ( log left(frac{boldsymbol{a}^{2}}{boldsymbol{a}^{2}-boldsymbol{v}^{2}}right), ) where ( boldsymbol{m} boldsymbol{g}=boldsymbol{k} boldsymbol{a}^{2} )
A. True
B. False
12
654 The differential equation of the simple harmonic motion given by ( boldsymbol{x}= )
( boldsymbol{A} cos (boldsymbol{n} boldsymbol{t}+boldsymbol{alpha}) ) is
( ^{A} cdot frac{d^{2} x}{d t^{2}}-n^{2} x=0 )
B. ( frac{d^{2} x}{d t^{2}}+n^{2} x=0 )
c. ( frac{d x}{d t}-frac{d^{2} x}{d t^{2}}=0 )
D. ( frac{d^{2} x}{d t^{2}}-frac{d x}{d t}+n x=0 )
12
655 Which one of the following is the differential equation that represents the family of curves ( y=frac{1}{2 x^{2}-c} ) where
( c ) is an arbitrary constant?
A ( cdot frac{d y}{d x}=4 x y^{2} )
B. ( frac{d y}{d x}=frac{1}{y} )
c. ( frac{d y}{d x}=x^{2} y )
D. ( frac{d y}{d x}=-4 x y^{2} )
12
656 The D.E whose signature is ( y= )
( C_{1} e^{3 x}+C_{2} e^{5 x} ) is:
A ( cdot y_{2}+2 y_{1}+15 y=0 )
в. ( y_{2}+8 y_{1}+15 y=0 )
c. ( y_{2}+8 y_{1}-15 y=0 )
D. ( y_{2}-8 y_{1}+15 y=0 )
12
657 Obtain differential equation from the
relation ( boldsymbol{A} boldsymbol{x}^{2}+boldsymbol{B} boldsymbol{y}^{2}=1, ) where ( boldsymbol{A} ) and ( mathbf{B} )
are constants
12
658 Solve :
( frac{d y}{d x}=x(2 log x+1), operatorname{given} y=0 ) where
( boldsymbol{x}=mathbf{2} )
12
659 The order and degree of the differential
equation of all parabola whose axis is ( x ) axis
( A cdot 2, )
B. 2,
( c cdot 1,2 )
D. 1,1
12
660 The differential coefficient of log (tan
( x ) )is
A ( .2 sec 2 x )
B. ( 2 cos e c 2 x )
( mathbf{c} cdot 2 sec ^{2} x )
D. ( 2 cos e c^{2} 2 x )
12
661 Determine the order and degree of the
following differential equation. State also whether it is linear or non-linear.
( left(y^{prime prime}right)^{2}+left(y^{prime}right)^{3}+sin y=0 )
12
662 Degree of ( left(frac{d y}{d x}right)^{2}+3 frac{d^{2} y}{d x^{2}}=sqrt{1+left(frac{d y}{d x}right)^{2}} )
is:
( A )
B. 2
( c cdot 3 )
( D )
12
663 Which of the following is true regarding the function ( boldsymbol{f}(boldsymbol{x}, boldsymbol{y})=boldsymbol{x}^{4} sin frac{boldsymbol{x}}{boldsymbol{y}} ? )
A. Not homogenous
B. Homogenous with degree 2
c. Homogenous with degree 3
D. None of the above
12
664 Consider the equation ( frac{x^{2}}{a^{2}+lambda}+ ) ( frac{y^{2}}{b^{2}+lambda}=1 ) where a and ( b ) are specified
constants and ( lambda ) is an arbitrary
parameter. Find a differential equation satisfied by it.
12
665 The D.E. whose solution is ( boldsymbol{y}=boldsymbol{c}_{1}+boldsymbol{c}_{2} boldsymbol{e}^{boldsymbol{x}} )
is:
( mathbf{A} cdot y^{prime prime}=y )
B . ( y^{prime prime}=y^{prime} )
c. ( y^{prime prime}+y=0 )
D cdot ( y^{prime prime}+y^{prime}=0 )
12
666 The order and degree of the differential
equation ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{3 / 2}=boldsymbol{y} ) are
respectively
A . 1,1
в. 1,2
( mathrm{c} .1,3 )
D. 2,
E .2,2
12
667 Find the integral curve of the differential equation, ( x(1-x ln y) cdot frac{d y}{d x}+y=0 ) which passes
hrough ( left(1, frac{1}{e}right), ) then ( (2+ln y) x ) is:
A . 1
B. 2
c. 0
D. None of these
12
668 Solve the differential equation ( boldsymbol{x} boldsymbol{d} boldsymbol{y}+ )
( 2 y d x=0 ) when ( x=2, y=1 )
12
669 The order, degree of the differential
equation satisfying the relation ( sqrt{1+x^{2}}+sqrt{1+y^{2}}=lambda(x sqrt{1+y^{2}}) )
( left.y sqrt{1}+x^{2}right) ) is
A . 1,1
в. 2,
c. 3,2
D. 0,1
12
670 16.
The population p (t) at time t of a certain mouse species
dp(t)
satisfies the differential equation –
-= 0.5 p(t) – 450.
dt
Ifp (0) = 850, then the time at which the population becomes
zero is :
[2012]
(a) 2ln 18
(b) In 9
(c)
= In 18 (d) In 18

metod
12
671 Find the order and degree of ( (1+ )
( left.boldsymbol{y}^{prime}right)^{1 / 2}=boldsymbol{y}^{prime prime} )
A .2,2
в. 1,1
c. 1,2
D. 2,
12
672 The order of the differential equation ( 2 x^{2} frac{d^{2} y}{d x^{2}}-3 frac{d y}{d x}+y=0 )
A . 2
B.
( c cdot 0 )
D. Not defined
12
673 If(a + bx) eYx=x, then prove that x
dx
(1983 – 3 Marks)
12
674 If ( boldsymbol{y}=boldsymbol{a} cos (log boldsymbol{x})+boldsymbol{b} sin (log boldsymbol{x}) ) then
prove that ( boldsymbol{x}^{2} frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{y}=boldsymbol{0} )
12
675 If ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}^{-3} ) then ( boldsymbol{y} )
A. ( -frac{1}{2 x^{2}}+c )
в. ( frac{1}{2 x^{2}}+c )
c. ( -frac{1}{3 x^{3}}+c )
D. ( -frac{1}{4 x^{4}}+c )
12
676 Solve ( : boldsymbol{y} log boldsymbol{y} frac{d boldsymbol{x}}{d boldsymbol{y}}+boldsymbol{x}-log boldsymbol{y}=mathbf{0} ) 12
677 Find the general solution of ( (x+ ) ( left.2 y^{3}right) frac{d y}{d x}=y ) 12
678 What is the general solution of the
differential equation ( x^{2} d y+y^{2} d x=0 ? )
A. ( x+y=c ) where ( c ) is the constant of integration
B. ( x y=c ) where ( c ) is the constant of integration
c. ( c(x+y)=x y ) where ( c ) is the constant of integration
D. None of the above
12
679 Consider the following equation, ( frac{d y}{d x}+ ) ( boldsymbol{P}(boldsymbol{x}) boldsymbol{y}=boldsymbol{Q}(boldsymbol{x}) )
(i) If two particular solutions of given equation ( u(x) ) and ( v(x) ) are known, find the general solution of the same equation in terms of ( u(x) ) and ( v(x) )
(ii) If ( alpha ) and ( beta ) are constants such that
the linear combinations ( boldsymbol{alpha} cdot boldsymbol{u}(boldsymbol{x})+boldsymbol{beta} )
( v(x) ) is a solution of the given equation,
find the relation between ( alpha ) and ( beta )
(iii) If ( boldsymbol{w}(boldsymbol{x}) ) is the third particular
solution different from ( u(x) ) and ( v(x) ) then find the ratio ( frac{boldsymbol{v}(boldsymbol{x})-boldsymbol{u}(boldsymbol{x})}{boldsymbol{w}(boldsymbol{x})-boldsymbol{u}(boldsymbol{x})} )
12
680 Obtain the differential equation of the family of circles touching the y-axis at the origin. 12
681 The solution of ( frac{d^{2} y}{d x^{2}}=x e^{x}+1 ) is:
A ( cdot y=(x-1) e^{x}+frac{1}{2} x^{2}+C_{1} x+C_{2} )
B. ( y=(x-2) e^{x}+frac{1}{2} x^{2}+C_{1} x+C_{2} )
c. ( y=(x+2) e^{x}+frac{1}{2} x^{2}+C_{1} x+C_{2} )
D. ( y=(x+2) e^{x}+C_{1} )
12
682 The order and power of differential
equation ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+boldsymbol{y} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+int boldsymbol{y} boldsymbol{d} boldsymbol{x}=sin boldsymbol{x} ) is
A . 1,3
в. 3,1
c. 1.2
D. 2,1
12
683 Example 2.1 A police jeep, approaching a right-angled
intersection from the north, is chasing a speeding car that has
turned the corner and is now moving straight east. When the
jeep is 0.6 km north of the intersection and the car is 0.8 km
to the east, the police determine with radar that the distance
between them and the car is increasing at 20 km h’. If the
jeep is moving at 60 km h at the instant of measurement,
what is the speed of the car?
12
684 The differential equation representing the family of curves
12 =2c(x+Vc), where c>0, is a parameter, is of order and
degree as follows:
[2005]
(a) order 1, degree 2 (b) order 1, degree 1
(c) order 1, degree 3 (d) order 2, degree 2
12
685 If ( (2+sin x) frac{d y}{d x}+(y+1) cos x=0 ) and
( y(0)=1, ) then ( y=left(frac{pi}{2}right) ) is equal to
A ( cdot frac{1}{3} )
B. ( -frac{2}{3} )
( c cdot-frac{1}{3} )
D.
12
686 Find the general solution of the
differential equation:
( (1+x)left(1+y^{2}right) d x+ )
( (1+y)left(1+x^{2}right) d y=0 )
12
687 ( frac{d^{2} y}{d x^{2}}+sin left(frac{d y}{d x}right)+y 0 ) Find order and
degree of this ( D E )
12
688 ( boldsymbol{y}-boldsymbol{x} frac{d boldsymbol{y}}{d boldsymbol{x}}=mathbf{5}left(boldsymbol{y}^{2}+frac{d boldsymbol{y}}{d x}right) ) 12
689 ( y+x^{2}=frac{d y}{d x} ) has the solution:
A ( cdot y+x^{2}+2 x+2=c cdot e^{x} )
B. ( y+x+2 x^{2}+2=c . e^{x} )
C ( cdot y+x+x^{2}+2=c cdot e^{2 x} )
D. ( y^{2}+x+x^{2}+2=c . e^{2 x} )
12
690 If the general solution of some
differential equation is ( boldsymbol{y}=boldsymbol{a}_{1}left(boldsymbol{a}_{2}+right. )
( left.boldsymbol{a}_{3}right) cdot cos left(boldsymbol{x}+boldsymbol{a}_{4}right)-boldsymbol{a}_{5} boldsymbol{e}^{x+boldsymbol{a}_{6}} quad ) then order
of differential equation is
( mathbf{A} cdot mathbf{6} )
B. 5
( c cdot 4 )
D. 3
12
691 Let ( y=y(x) ) be the solution of the differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{2} boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) )
where ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}mathbf{1}, & boldsymbol{x} in[mathbf{0}, mathbf{1}] \ mathbf{0}, & text { otherwise }end{array}right. )
If ( boldsymbol{y}(mathbf{0})=mathbf{0}, ) then ( boldsymbol{y}left(frac{mathbf{3}}{mathbf{2}}right) ) is
( ^{A} cdot frac{e^{2}-1}{2 e^{3}} )
в. ( frac{e^{2}-1}{e^{3}} )
c. ( frac{1}{2 e} )
D. ( frac{e^{2}+1}{2 e^{4}} )
12
692 Degree and order of the differential equation ( frac{d^{2} y}{d x^{2}}=left(frac{d y}{d x}right)^{2} ) are
respectively
A. 1,2
в. 2,
( c .2,2 )
D. ( 1, )
12
693 Let ( y=f(x) ) be the solution of ( frac{d y}{d x}=frac{y}{x}+ ) ( tan frac{boldsymbol{y}}{boldsymbol{x}}, boldsymbol{y}(1)=boldsymbol{pi} / 2 ) then 12
694 A certain radioactive material is known
to decay at a rate proportional to the amount present. If after one hour it is
observed that 10 percent of the material has decayed, find the half-life (period of time it takes for the amount of material
to decrease by half) of the material (in hrs.)
A . 6.58
B. 8.58
c. 10.58
D . 12.58
12
695 The degree and order of the differential equation of the family of all parabolas whose axis is ( x ) -axis are respectively:
A . 2,1
в. 1,2
( c .3,2 )
D. 2,3
12
696 If ( boldsymbol{y}=log left(1+2 t^{2}+boldsymbol{t}^{4}right), boldsymbol{x}=tan ^{-1} boldsymbol{t} )
find ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} )
12
697 Define particular solution of a differential equation 12
698 For each of the differential equations
given in exercises 1 to 12 .find the general solution. ( frac{d y}{d x}+2 y=sin x )
12
699 Show that the function of ( boldsymbol{y}= )
( A sin 2 x+B cos 2 x ) satisfies the
differential equation ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+boldsymbol{4} boldsymbol{y}=mathbf{0} )
12
700 Find a particular solution of the differential equation ( frac{d y}{d x}+y cot x= ) ( 4 x operatorname{cosec} x(x neq 0), ) given that ( y=0 )
when ( boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{2}} )
12
701 10. If y = y(x) satisfies the differential equation
87# (V9+ Vx)dy = (14+ 10 + ve) dx, x>0 and
y(0)= 57, then y (256)=
(JEE Adv. 2018)
(a) 3
(b) 9
(c) 16
(d) 80
12
702 The differential equation whose
solution is ( A x^{2}+B y^{2}=1, ) where ( A ) and
B are arbitrary constants is of
A. second order and second degree
B. first order and second degree
c. first order and first degree
D. second order and first degree
12
703 If the population of a country doubles in
60 years, in how many years will it be triple under the assumption that the rate of increase is proportional to the number of inhabitants?
( [text { Given: } log 2=0.6912 text { and } log 3= )
1.0986.
12
704 3.
ugh (1,1) and at P(x, y), tangent
and B respectively such that
(2006 – 5M, -1)
A curve y=f(x) passes through (1, 1) and at
cuts the x-axis and y-axis at A and B respectively su
BP: AP=3:1, then
(a) equation of curve is xy’ – 3y=0
(b) normal at(1, 1) is x + 3y=4
(C) curve passes through (2, 1/8)
(d) equation of curve is xy’ + 3y=0
12
705 The differential equation of the family of parabolas with vertex at (0,-1) and having axis along the ( y ) -axis is:
A ( cdot y y^{prime}+2 x y+1=0 )
В . ( x y^{prime}+y+1=0 )
c. ( x y^{prime}+2 y+2=0 )
D. ( x y^{prime}-y-1=0 )
12
706 The differential equation corresponding
to the family of circles given by
( (x-a)^{2}+(y-b)^{2}=4 ) where a and
are parameters, is
A ( cdot 4 frac{d^{2} y}{d x^{2}}+9 y=0 )
( 4left(frac{d^{2} y}{d x^{2}}right)^{2}=left[1+left(frac{d y}{d x}right)^{2}right]^{3} )
( ^{mathbf{C}} 4 frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}=6 y )
( 4left(frac{d^{2} y}{a x^{2}}right)^{2}+left[1+left(frac{d y}{d x}right)^{2}right]^{2}=0 )
12
707 The ( D . E ) of the family of circles
touching ( X- ) axis at (0,0) is
The ( D . E ) of the family of circle passing through the origin and having their centre on ( boldsymbol{Y}-boldsymbol{a} boldsymbol{x} boldsymbol{i} boldsymbol{s} ) is
A ( cdot frac{d y}{d x}=frac{2 x y}{x^{2}-y^{2}} )
в. ( frac{d y}{d x}=frac{x^{2}-y^{2}}{2 x y} )
c. ( frac{d y}{d x}=x^{2}-y^{2} )
D. ( frac{d y}{d x}+frac{x}{y}=0 )
12
708 Solve the following differential equation ( frac{d y}{d x}=sec ^{2} x ) 12
709 The solution of the equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( frac{y}{x}left(log frac{y}{x}+1right) ) is
( ^{A} cdot log frac{y}{x}=c x )
в. ( frac{y}{x}=log y+c )
c. ( y=log y+1 )
D. ( y=x y+c )
12
710 ( boldsymbol{x}(boldsymbol{x}-1) frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}-(boldsymbol{x}-boldsymbol{2}) boldsymbol{y}=boldsymbol{x}^{3}(boldsymbol{2} boldsymbol{x}- )
1). The solution to the above given differential equation is ( boldsymbol{y}(boldsymbol{x}+boldsymbol{k})= )
( boldsymbol{x}^{m}left(boldsymbol{x}^{n}-boldsymbol{x}+boldsymbol{c}right) . ) Find ( boldsymbol{k}+boldsymbol{m}+boldsymbol{n} ? )
12
711 The initial mass of a radioactive
isotope was 128 g. Find the mass of a material if 4 half lives occured.
( A cdot 8 g )
в. 7.5
( c .7 g )
D. ( 6.5 mathrm{g} )
12
712 Construct a differential equation by eliminating the arbitrary constants
( A, B, C ) for the equation ( y^{2}=A x^{2}+ )
( boldsymbol{B} boldsymbol{x}+boldsymbol{C} )
12
713 The differential equation whose solution is ( (boldsymbol{x}-boldsymbol{h})^{2}+(boldsymbol{y}-boldsymbol{k})^{2}=boldsymbol{a}^{2} ) is
( (a text { is a constant }) )
( ^{mathbf{A}} cdotleft[1+left(frac{d y}{d x}right)^{2}right]^{3}=a^{2} frac{d^{2} y}{d x^{2}} )
B ( cdotleft[1+left(frac{d y}{d x}right)^{2}right]^{3}=a^{2}left(frac{d^{2} y}{d x^{2}}right)^{2} )
( ^{mathbf{c}} cdotleft[1+left(frac{d y}{d x}right)right]^{3}=a^{2}left(frac{d^{2} y}{d x^{2}}right)^{2} )
D. None of these
12
714 The order and degree of the differential equation ( sqrt{frac{boldsymbol{d y}}{boldsymbol{d} boldsymbol{x}}}-4 frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}-boldsymbol{7} boldsymbol{x}=boldsymbol{0} ) are
A. 1 and ( 1 / 2 )
B. 2 and 1
c. 1 and 1
D. 1 and 2
12
715 The differential equation of a free falling body is governed by the DE ( v frac{d v}{d x}= ) ( -c x-b v^{2} ).where ( v ) and ( x ) are velocity
and displacement respectively. The velocity of the body is given by the equation ( v^{2}=frac{c}{2 b^{2}}left(1-e^{-2 b x}right)-frac{c x}{b} )
A. True
B. False
12
716 The solution of ( y d x+x d y=d x+d y ) is:
A. ( x y=x+y+c )
В ( cdot x-y frac{x}{y}+c=0 )
c. ( x y-x+y=c )
D. ( x+y frac{x}{y}+c=0 )
12
717 The equation of the curve through ( left(0, frac{pi}{4}right) ) satisfying the differential
equation. ( e^{x} tan y d x+(1+ )
( left.e^{x}right) sec ^{2} y d y=0 ) is given by
A ( cdotleft(1+e^{x}right) tan y=2 )
B . ( 1+e^{x}=2 tan y )
C ( cdot 1+e^{x}=2 sec y )
D・ ( left(1+e^{x}right) tan y=1 )
12
718 Distinguish which one is initial valued
ordinary differential equation and
boundary valued ordinary differentia equation:
( mathbf{y}^{prime prime}+mathbf{2} boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}}, boldsymbol{y}(boldsymbol{pi})=mathbf{1} ; boldsymbol{y}^{prime}(boldsymbol{pi})=mathbf{2} )
¡) ( boldsymbol{y}^{prime prime}+mathbf{2} boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}}, boldsymbol{y}(mathbf{0})=mathbf{1} ; boldsymbol{y}^{prime}(mathbf{1})=mathbf{1} )
12
719 dy
24.
The solution of the differential equation x

+ 2y = x2
dx
(x+0) with y(1)=1, is:
JEEM 2019-9 April (M)
4
3 +
(a) y= 5* + 5×2
(b) y= 5 + 5x?
12
720 Solve: ( (boldsymbol{x}-boldsymbol{y} ln boldsymbol{y}+boldsymbol{y} ln boldsymbol{x}) boldsymbol{d} boldsymbol{x}+ )
( boldsymbol{x}(ln boldsymbol{y}-ln boldsymbol{x}) boldsymbol{d} boldsymbol{y}=mathbf{0} )
A ( cdot x ln left(frac{y}{x}right)-y+x ln x+c y=0 )
в. ( y ln left(frac{x}{y}right)-y+x ln x+c x=0 )
c. ( x ln left(frac{x}{y}right)-y+x ln x+c y=0 )
D. ( y ln left(frac{y}{x}right)-y+x ln x+c x=0 )
12
721 The curve amongst the family of curves, represented by the differential
equation, ( left(x^{2}-y^{2}right) d x+2 x y d y=0 )
which passes through (1,1) is :
A. A circle with centre on the y-axis
B. A circle with centre on the x-axis
c. An ellipse with major axis along the y-axis
D. A hyperbola with transverse axis along the x-axis
12
722 Assertion
The differential equation of all circles in
a plane is of order 3
Reason
Only one circle can be drawn through any three points in a plane
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
723 ff ( boldsymbol{y}=boldsymbol{A} boldsymbol{e}^{-boldsymbol{k} t} cos (boldsymbol{p} boldsymbol{t}+boldsymbol{c}), ) then prove
that ( frac{d^{2} y}{d t^{2}}+2 k frac{d y}{d t}+n^{2} y=0, ) where
( boldsymbol{n}^{2}=boldsymbol{p}^{2}+boldsymbol{k}^{2} )
12
724 The time after which the cone is empty
is
( mathbf{A} cdot H / 2 k )
в. ( H / k )
( mathbf{c} cdot H / 3 k )
D. ( 2 H / k )
12
725 If ( x=sin ^{-1} 2 t sqrt{1-t^{2}} ) and ( y=frac{pi}{2} )
( cos ^{-1} t, ) then find the value of ( frac{d^{2} y}{d x^{2}} ) at ( t=frac{pi}{3} )
12
726 Find the half life of a radioactive
element, if its activity decreases for 1 month by ( 10 % )
A. 193.3 days
B. 197.3 days
c. 198.5 days
D. 199.7 days
12
727 Solve: ( sin x frac{d y}{d x}-y=sin x cdot tan frac{x}{2} ) 12
728 Lt is known that the decay rate of radium is directly proportional to its quantity at each given instant Find the law of variation of a mass of radium
as a function of time if at ( t=0, ) the
mass of the radius was ( m_{0} ) and
during time ( t_{0} alpha % ) of the original mass of radium decay, if ( m=m_{0} e^{-k t}, ) then
( k=? )
A ( cdot k=frac{1}{t_{0}} ln left(1-frac{alpha}{100}right) )
В ( cdot k=frac{-1}{t_{0}} ln left(1-frac{alpha}{100}right) )
C ( cdot k=frac{1}{t_{0}} ln (1-alpha) )
D・ ( k=frac{-1}{t_{0}} ln (1-alpha) )
12
729 Find the solution of
( (3 x+4 y-5)^{2} frac{d y}{d x}=a^{2} )
A ( quad 4 y+lambda=frac{2 a}{sqrt{3}} tan ^{-1} frac{(3 x+4 y-5) sqrt{3}}{2 a} )
B. ( 2 y+lambda=frac{5 a}{sqrt{3}} tan ^{-1} frac{(3 x+4 y-5) sqrt{3}}{2 a} )
c. ( quad 2 y+lambda=frac{2 a}{sqrt{3}} tan ^{-1} frac{(3 x+4 y-5) sqrt{3}}{2 a} )
D. ( 4 y+lambda=frac{5 a}{sqrt{3}} tan ^{-1} frac{(3 x+4 y-5) sqrt{3}}{2 a} )
12
730 Solve :
( log left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)=boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{y} )
12
731 Solve the differential equation:
( boldsymbol{y} log boldsymbol{y} frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{d} boldsymbol{y}}+boldsymbol{x}-log boldsymbol{y}=mathbf{0} )
12
732 The solution of differential equation
( left(e^{x}+1right) y d y=(y+1) e^{x} d x ) is
A ( cdotleft(e^{x}+1right)(y+1)=C e^{y} )
B . ( left(e^{x}+1right)|(y+1)|=C e^{-y} )
C ( cdotleft(e^{x}+1right)(y+1)=pm C e^{y} )
D. None of these
12
733 It is known that the decay rate of radium is directly proportional to its quantity at each given instant. Find the law of variation of a mass of radium as
a function of time if at ( t=0, ) the mass of
the radius was ( m_{0} ) and during time ( t_{0} alpha )
( % ) of the original mass of radium decay.
A ( quad m=m_{0} e^{-k t} ) where ( k=frac{1}{t_{0}} ln left(1-frac{alpha}{100}right) )
B. ( m=m_{0} e^{-k(t-1)} ) where ( k=frac{1}{t_{0}} ln left(1+frac{alpha}{100}right) )
c. ( m=m_{0} e^{-k(t-1)} ) where ( k=frac{1}{t_{0}} ln left(1-frac{alpha}{100}right) )
D. ( m=m_{0} e^{-k t} ) where ( k=frac{1}{t_{0}} ln left(1+frac{alpha}{100}right) )
12
734 Find the differential equation of the
family of curves ( boldsymbol{y}=boldsymbol{A} boldsymbol{e}^{2 boldsymbol{x}}+boldsymbol{B} boldsymbol{e}^{-boldsymbol{2 x}} )
where ( A ) and ( B ) are arbitrary constants.
12
735 Solve the differential equation:
( boldsymbol{y} log boldsymbol{y} boldsymbol{d} boldsymbol{x}-boldsymbol{x} boldsymbol{d} boldsymbol{y}=mathbf{0} )
12
736 Obtain a differential equation by
eliminating the arbitrary constants ( a )
and ( b ) from the equation ( y=a cos n t+ )
( boldsymbol{b} sin boldsymbol{n} boldsymbol{t} )
12
737 The ( D . E ) of the family of concentric
circles with centre at origin is
A. ( x d x+y d x=0 )
B. ( x=y frac{d y}{d x} )
c. ( frac{d y}{d x}=frac{y}{x} )
D. ( frac{d y}{d x}=frac{x^{2}}{y^{2}} )
12
738 Solve the following differential equation ( frac{d y}{d x}=e^{x} ) 12
739 Solve: ( frac{d y}{d x}+2 y tan x=sin x ) 12
740 ff ( y=y(x) ) satisfies the differential
equation
[
begin{array}{c}
mathbf{8} sqrt{boldsymbol{x}}(sqrt{mathbf{9}+sqrt{boldsymbol{x}}}) boldsymbol{d} boldsymbol{y}= \
(sqrt{4+sqrt{mathbf{9}+sqrt{boldsymbol{x}}}})^{-1} boldsymbol{d} boldsymbol{x}, boldsymbol{x}>mathbf{0} text { and } \
boldsymbol{y}(mathbf{0})=sqrt{mathbf{7}}, text { then } boldsymbol{y}(mathbf{2 5 6})_{—}
end{array}
]
A . 16
B. 80
( c .3 )
( D )
12
741 Solve the following differential equation ( frac{d y}{d x}=x^{2} ) 12
742 10.
A curve ‘C’ passes through (2,0) and the slope at (x, y) as
2. Find the equation of the curve. Find the
x +1
area bounded by curve and x-axis in fourth quadrant.
12
743 Write the integrating factor of the following differential equation:
( left(1+y^{2}right)+(2 x y-cot y) frac{d y}{d x}=0 )
12
744 State whether the following statement
is True or False.
The current in the circuit with
inductance ( L ) and resistance ( R ) and
voltage ( boldsymbol{E} sin omega boldsymbol{t} ) is given by ( boldsymbol{L} frac{boldsymbol{d} boldsymbol{i}}{boldsymbol{d} boldsymbol{t}}+boldsymbol{R} boldsymbol{i}= )
Esinwt.if ( i=0 ) at ( t=0 ) then,current is
given by ( i=frac{E}{sqrt{E^{2}+L^{2} omega^{2}}}[sin (omega t- )
( left.phi)+e^{-R t / L} sin phiright] ) where ( phi= )
( tan ^{-1}left(frac{L omega}{R}right) )
A. True
B. False
12
745 The temperature of a body decreased
from ( 200^{circ} ) to ( 100^{circ} ) in 1 hour. Determine
how many degrees the body cooled in one hour more if the environment
temperature is ( 0^{circ} ? )
A ( .30^{circ} )
B . ( 40^{circ} )
( c cdot 50^{circ} )
D. ( 60^{circ} )
12
746 Order of ( left(frac{d y}{d x}+3 xright)^{3 / 2}=x+frac{3 d y}{d x} ) is:
( A cdot 3 )
B. 2
( c )
( D )
12
747 3.
The degree and order of the differential equation of the
family of all parabolas whose axis is x-axis, are res
[2003]
(2) 23 (6) 2,1 (c) 1,2 (d) 3,2.
12
748 The solution of the differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{y}}{boldsymbol{x}}+frac{boldsymbol{varphi}(boldsymbol{y} / boldsymbol{x})}{boldsymbol{varphi}^{prime}(boldsymbol{y} / boldsymbol{x})} ) is :
A ( cdot x varphileft(frac{y}{x}right)=k )
в. ( k varphileft(frac{y}{x}right)=x )
c. ( operatorname{kyvarphi}left(frac{y}{x}right)=y )
D ( cdot varphileft(frac{y}{x}right)=k y )
12
749 The normal at any point ( boldsymbol{P}(boldsymbol{x}, boldsymbol{y}) ) of ( mathbf{a} )
curve meets the ( x ) -axis at ( Q ) and ( N ) is
the foot of the ordinate at ( P )

If ( N Q=frac{xleft(1+y^{2}right)}{1+x^{2}}, ) then equation of such curve, given that it passes through the point (3,1) is:
A. ( x^{2}-y^{2}=8 )
B . ( x^{2}+2 y^{2}=11 )
c. ( x^{2}-5 y^{2}=4 )
D. ( x^{2}+3 y^{2}=7 )

12
750 The differential equation of the family of straight lines which passes through the origin is
A ( cdot y=x frac{d y}{d x} )
в. ( y+x frac{d y}{d x}=0 )
c. ( _{x+y} frac{d y}{d x}=0 )
D. ( frac{d y}{d x}=m )
12
751 ff ( boldsymbol{y}=boldsymbol{y}(boldsymbol{x}) ) satisfies the differential equation ( 8 sqrt{x}(sqrt{9+sqrt{x}}) d y= )
( (sqrt{4+sqrt{9+sqrt{x}}})^{-1} d x, x>0 ) and
( boldsymbol{y}(mathbf{0})=sqrt{mathbf{7}}, ) then ( boldsymbol{y}(mathbf{2 5 6})= )
( A )
B.
c. 16
D. 80
12
752 Assertion
The order of the differential equation, of
which ( boldsymbol{x} boldsymbol{y}=boldsymbol{c} boldsymbol{e}^{boldsymbol{x}}+boldsymbol{b} boldsymbol{e}^{-boldsymbol{x}}+boldsymbol{x}^{boldsymbol{2}} ) is a
solution, is 2
Reason
The differential equation is ( x frac{d^{2} y}{d x^{2}}+ ) ( 2 frac{d y}{d x}-x y+x^{2}-2=0 )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
753 Find the order and degree of the differential equations.
a) ( y^{prime prime}+3 y^{prime}^{2}+y^{3}=0 )
b) ( left(frac{d y}{d x}right)^{2}+frac{1}{d y d x}=2 )
c) ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+boldsymbol{4} boldsymbol{y}=mathbf{0} )
12
754 Write the order and degree of the differential equation ( boldsymbol{y}=boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+ )
( boldsymbol{a} sqrt{1+left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}} )
12
755 Solve the following differential equation ( cos x frac{d y}{d x}-cos 2 x=cos 3 x ) 12
756 The D.E whose solution is ( boldsymbol{y}= )
( a cos (3 x+b) ) is?
A ( cdot y_{2}+3 y=0 )
В. ( y_{2}+y=0 )
c. ( y_{2}+9 y=0 )
D. ( y_{2}+6 y=0 )
12
757 If ( x^{y}=e^{x-y}, ) show that ( frac{d y}{d x}= ) ( frac{boldsymbol{y} log boldsymbol{x}}{boldsymbol{x}(log boldsymbol{x}+mathbf{1})} ) 12
758 Degree of ( left[frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}right]^{3 / 2}=k frac{d^{2} y}{d x^{2}} )
is:
( A cdot 4 )
B.
( c cdot 2 )
( D )
12
759 The degree of the differential equation ( 1+left(frac{d y}{d x}right)^{2}=x ) is 12
760 Which of the following is/are correct
regarding homogeneous differential
equation?
This question has multiple correct options
A. Represented in the form: ( M(x, y) d x-N(x, y) d y=0 )
B. Represented in the form: ( M(x, y) d x+N(x, y) d y=0 )
C. Both ( M(x, y), N(x, y) ) are homogeneous functions of the same degree
D. ( M(x, y), N(x, y) ) are homogeneous functions with different degrees
12
761 Solve the differential equation: ( frac{boldsymbol{d} z}{boldsymbol{d} boldsymbol{x}}+ ) ( frac{z}{x} log z=frac{z}{x^{2}}(log z)^{2} )
A ( cdot frac{1}{-x log z}=frac{1}{2 x^{2}}-c )
в. ( frac{1}{x log z}=frac{1}{2 x^{-2}}-c )
c. ( frac{1}{x log z}=frac{-1}{2 x^{2}}-c )
D. ( frac{1}{x log z}=frac{1}{2 x^{2}}-c )
12
762 An integrating factor of the differential
equation ( x d y-y d x+x^{2} e^{x} d x=0 ) is
A ( cdot frac{1}{x} )
B. ( log sqrt{1+x^{2}} )
c. ( sqrt{1+x^{2}} )
D.
E ( frac{1}{1+x^{2}} )
12
763 Solve:
( x d y+y d x=x^{2} y d y )
12
764 If ( (x-a)^{2}+(y-b)^{2}=c^{2}, ) then prove
that ( frac{left[1+left(frac{d y}{d x}right)^{2}right]^{3 / 2}}{frac{d^{2} y}{d x^{2}}} ) is a independent
of ( boldsymbol{C} )
12
765 For the differential equation ( x frac{d y}{d x}- ) ( boldsymbol{y}=sqrt{left(boldsymbol{x}^{2}+boldsymbol{y}^{2}right)}, ) show that its solution
is ( y+sqrt{left(x^{2}+y^{2}right)}=k x^{2} )
12
766 If ( boldsymbol{x}=sin t, boldsymbol{y}=cos boldsymbol{p} boldsymbol{t}, ) then
B . ( left(1-x^{2}right) y_{2}+x y_{1}-p^{2} y=0 )
C ( cdotleft(1+x^{2}right) y_{2}-x y_{1}+p^{2} y=0 )
D. ( left(1-x^{2}right) y_{2}-x y_{1}+p^{2} y=0 )
12
767 The D.E. obtained from ( y=c x^{2}+c^{3} ) is
( ^{mathbf{A}} cdotleft(frac{d y}{d x}right)^{3}+4 x^{4} frac{d y}{d x}=8 x^{3} y )
( ^{mathbf{B}}left(frac{d y}{d x}right)^{3}-4 x^{4} frac{d y}{d x}=8 x^{3} y )
( ^{mathrm{c}}-left(frac{d y}{d x}right)^{3}-4 x^{4} frac{d y}{d x}=8 x^{3} y )
( left(frac{d y}{d x}right)^{3}-4 x^{4} frac{d y}{d x}=-8 x^{3} y )
12
768 Form the differential equation
representing the family of curves ( boldsymbol{y}= ) ( a sin (x+b), ) where ( a, b ) are arbitrary
constants.
12
769 The differential equation which
represents the family of curves ( mathbf{y}= )
( mathbf{c}_{1} mathbf{e}^{mathbf{c}_{2} mathbf{x}} ) where ( mathbf{c}_{1} ) and ( mathbf{c}_{2} ) are arbitrary
constants, is:
A ( cdot y^{prime}=y^{2} )
B. ( mathrm{y}^{prime prime}=mathrm{y}^{prime} mathrm{y} )
( mathbf{C} cdot mathbf{y y}^{prime prime}=mathbf{y}^{prime} )
D・yy” = (y’) 2
12
770 The rate at which the population of a city increases at any time is proportional to the population at that time. If there are 1,30,000 people in the city in 1960 and 1,60,000 in ( 1990, ) what approximate population may be anticipated in ( 2020 ? ) ( left[log _{e} frac{16}{13}=0.2070, e^{0.42}=1.52right] ) 12
771 If ( y=a sin x+b cos x, ) then ( y^{2}+ )
( left(frac{d y}{d x}right)^{2} ) is a
A. Function of ( x )
B. Function of ( y )
c. Function of ( x ) and ( y )
D. constant
12
772 If ( boldsymbol{y}=left(tan ^{-1} boldsymbol{x}right)^{2} )
S.T ( left(boldsymbol{x}^{2}+mathbf{1}right)^{2} boldsymbol{y}_{2}+boldsymbol{2} boldsymbol{x}left(boldsymbol{x}^{2}+mathbf{1}right) boldsymbol{y}_{1}=mathbf{2} )
12
773 Let ( boldsymbol{f}:[mathbf{1}, mathbf{3}] rightarrow mathbf{R} ) be a continuous
function that is differentiable in ( (mathbf{1}, mathbf{3}) )
and ( f^{prime}(x)=|f(x)|^{2}+4 ) for all ( x in(1,3) )
Then?
This question has multiple correct options
A ( . f(3)-f(1)=5 ) is true
B. ( f(3)-f(1)=5 ) is false
c. ( f(3)-f(1)=7 ) is false
D. ( f(3)-f(1)<0 ) only at one point of (1,3)
12
774 The slope of a curve at each of its points is equal to the square of the abscissae of the point. Find the particular curve
through the point (-1,1) ( mathbf{1}+boldsymbol{y}=boldsymbol{2} boldsymbol{e}^{boldsymbol{x}^{2} / mathbf{2}} )
12
775 What is the degree of the differential eqaution ( : frac{d^{3} y}{d x^{3}}-6left(frac{d y}{d x}right)^{2}-4 y=0 )
( A )
B. 2
( c cdot 3 )
D. None of these
12
776 Write the degree of the differential
equation ( a^{2} frac{d^{2} y}{d x^{2}}=left{1+left(frac{d y}{d x}right)^{2}right}^{1 / 4} )
12
777 Find the solution of ( left(frac{boldsymbol{x}+boldsymbol{y}-boldsymbol{a}}{boldsymbol{x}+boldsymbol{y}-boldsymbol{b}}right) frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}+boldsymbol{y}+boldsymbol{a}}{boldsymbol{x}+boldsymbol{y}+boldsymbol{b}} )
( ^{mathbf{A}} cdot(b-a) log left{(x+y)^{2}-a bright}=2(x-y)+k )
в. ( (b-a) log left{(x-y)^{2}-a bright}=2(x+y)+k )
c. ( (b-a) log left{(x-2 y)^{2}-a bright}=2(x-y)+k )
D. ( (b-a) log left{(x+y)^{2}-a bright}=2(x+2 y)+k )
12
778 Form the differential equation from the following primitives, where constant is arbitrary. ( boldsymbol{y}=boldsymbol{c} boldsymbol{x}+boldsymbol{2} boldsymbol{c}^{2}+boldsymbol{c}^{boldsymbol{3}} ) 12
779 Degree of ( boldsymbol{y} frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}=left[boldsymbol{3}+left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{2}right]^{frac{2}{3}} ) is:
( A cdot 4 )
в. 3
( c cdot 2 )
( D )
12

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