# 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 NoQuestionsClass
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
2Eliminate 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
3A 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
4The 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
5Solution 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
6The 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
7The 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
8The 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
9Find 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
10The 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
11Find 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
12Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) if ( boldsymbol{y}=boldsymbol{x}^{2}-boldsymbol{2}^{sin boldsymbol{x}} )12
13The 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
1421. If(2+ sinx)
+ (y+1) cos x=0 and y(0) = 1, then
dx
is equal to :
[JEE M 2017
– /*
ها را با دی
/*
12
15Solve:
¡) ( 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
16The 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
17Find the solution of the differential
equation ( (x log x) frac{d y}{d x}+y= )
( 2 x log x,(x geq 1) )
12
18The 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
19Solve 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
20The differential equation whose solution is Ax? + By = 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
21Find the differential equation of ( boldsymbol{y}= )
( boldsymbol{a} e^{3 x}+boldsymbol{b} e^{3 x} )
12
22The 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
23The 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
24Solve the following differential equation ( frac{d y}{d x}=1-cos x )12
25Evaluate ( left(1-x^{2}right) frac{d y}{d x}-x y=1 )12
26Find the differential equation of the family of curve represented by
( c(y+c)^{2}+x^{3}=0 )
12
27The 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
28the 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
29The 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
31The 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
32The number of arbitrary constant in the particular solution of a differential equation is
( A cdot 3 )
B. 4
c. infinite
D. zero
12
33The 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
34If ( 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
35The 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
36The 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
37Solve 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
38Form the differential equation of the family of circles touching the X-axis at the origin.12
39Solve 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
40The 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
41Solve 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
1. ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}}+mathbf{1} quad: boldsymbol{y}^{prime prime}-boldsymbol{y}=mathbf{0} )
( mathbf{2}=mathbf{0} )
( sin x=0 )
4. ( y=sqrt{1+x^{2}} )
( : boldsymbol{y}^{prime}= )
( frac{x y}{1+x^{2}} )
( 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}) )
( 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
43The order of the differential equation is
( A cdot 3 )
B. 2
( c cdot 1 )
D. None of these
12
44Which 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
45Solve the differential equation ( boldsymbol{x}+ )
( y frac{d y}{d x}=2 y )
12
46solve
( x d y-y d x=sqrt{x^{2}+y^{2} d x} )
12
47Solution 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
48Solve: ( boldsymbol{y} boldsymbol{d} boldsymbol{x}+left(boldsymbol{x}-boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{y}=mathbf{0} )12
49The general solution of differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}+boldsymbol{y}}{boldsymbol{x}-boldsymbol{y}} ) is12
50Determine 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
51The 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
53If ( 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
54Find 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
55Solve:
( (mathbf{1}-boldsymbol{y}) boldsymbol{x} frac{d boldsymbol{y}}{d boldsymbol{x}}+(mathbf{1}+boldsymbol{x}) boldsymbol{y}=mathbf{0} )
12
56The 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
57Solve:
( frac{d y}{d x}=e^{4 x-3 y} )
12
58The 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
59The 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
60Solve ( boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{2} boldsymbol{y}=boldsymbol{x}^{2} log boldsymbol{x} )12
61I.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
62Form the differential equation of the
family of curves represented by ( y^{2}= )
( (x-c)^{3} )
12
635.
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
COs ec
e) sin() = log x + 6) cose()=logx +2
@sec ) 10g x +2 (a) cos(% )= log x +
12
64The 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
65The 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
669.
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
67Solve 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
68Solve the differential equation: ( boldsymbol{y} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( frac{boldsymbol{x}}{boldsymbol{e}^{boldsymbol{y}}} )
12
69If ( 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
70The 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
71The 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
72ntial 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
73Find 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
74Find the degree of each algebraic
expression
( 2 y^{2} z+10 y z )
12
75Assertion
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
76ff ( y=e^{a x}, ) then show that ( x frac{d y}{d x}=y log y )12
77It 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
78Solve: ( 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
79The 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
80Write 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
81The 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
82For the following differential equation, find the general solution.
( frac{d y}{d x}+x=1 )
12
83Solve:
( 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
84Write the differential equation
representing the family of curves ( boldsymbol{y}= )
( m x, ) where ( m ) is an arbitrary constant.
12
85The 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
86dy
– 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
87The 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
88The 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
89The 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
90Solve the following differential equation ( left(x^{2}+1right) frac{d y}{d x}=1 )12
91Find 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
92Solve 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
93The 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
94Solve 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
95Consider 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
96The 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
97Find 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
98The 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
99Let ( 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
100The 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
101Solution 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
102The 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
103The 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
104If 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
105Show that ( y=frac{1}{x} ) is a solution of the differential equation ( frac{d y}{d x}=log x )12
106The 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
107Show 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
108The number of people having flu after 20 days is
A. 1400
B. 1540
( c .1498 )
D. 1492
12
109The 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
110The 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
111If ( 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
112The 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
113Represent the following families of curves by forming the corresponding differential equation(a, b being parameters). ( x^{2}+y^{2}=a^{2} )12
114The 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
115D.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
116If 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
117Solution 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
118The 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
119A solution of the differential equation ( left(frac{d y}{d x}right)-x frac{d y}{d x}+y=0 ) is12
120The 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
121The 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
122The 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} ) are12
123The 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
125Find 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
126The 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
127D.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
128The 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
129Solve 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
130Find 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
131Solve: ( cos ^{2} x frac{d y}{d x}+y=tan x )12
132The 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
133Find 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
134The order and degree of the differential equation

(a) (1,5)
(C) (3,3)
(b) (3,1)
(d) (1,2)
12
135If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sin boldsymbol{x}, ) find ( boldsymbol{f}^{prime}(boldsymbol{pi}), ) using first
principle.
12
136Find the differential equation of ( y=A ) ( operatorname{cox}+B sin x )12
137Let ( _{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
138Solution 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
139Solve 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
140If ( 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
141Integrating factor of ( frac{boldsymbol{x} boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}-boldsymbol{y}=boldsymbol{x}^{4}-boldsymbol{3} boldsymbol{x} )
is:
12
142Consider 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
144If ( 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
145Form the differential equation corresponding to ( boldsymbol{x} boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+frac{boldsymbol{b}}{boldsymbol{x}},(boldsymbol{a}, boldsymbol{b}) )12
146Solve the differential equation ( frac{d y}{d x}+ )
( boldsymbol{y}=boldsymbol{e}^{-boldsymbol{x}} )
12
147Solve the following differential equation ( frac{d y}{d x}+1=sin x )12
148The 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
149The 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
150The 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
151Determine 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
152Show that ( 3 e^{x} tan y d x+(1- )
( left.e^{x}right) sec ^{2} y d y=0 )
12
153If ( 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
154Find 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
155Solve 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
156Solve 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
157The 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
158Find 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
1597.
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
160The 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
161Solve ( frac{d y}{d x}=cos (x+y) )
Solve ( frac{d y}{d x}=cos (x+y) )
12
162Solve:
( frac{cos ^{2} y}{x} d y+frac{cos ^{2} x d x}{y}=0 )
12
163Solve :
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=left(boldsymbol{1}+boldsymbol{x}^{2}right)left(boldsymbol{1}+boldsymbol{y}^{2}right) )
12
164If ( 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
165The 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
1661.
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
167The 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
168Form the differential equation
corresponding to ( boldsymbol{y}=boldsymbol{a} cos (boldsymbol{n} boldsymbol{x}+ )
( boldsymbol{b}),(boldsymbol{a}, boldsymbol{b}) )
12
169Find 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
170Find differential equation of all circles in the first quadrant which touch the co-ordinate axis.12
171f ( 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
172The 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
17320
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

(a) 1-k!
(b) 1- k(T – 1)2
2
(c) e-kł
(d) T² – 1
12
174Solve ( left(boldsymbol{y}+mathbf{3} boldsymbol{x}^{2}right) frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{d} boldsymbol{y}}=boldsymbol{x} )12
175Equation 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
176Solve ( 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
178The solution of ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x} boldsymbol{y}}{boldsymbol{x}^{2}+boldsymbol{y}^{2}} ) is
[
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
179Find 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
180Find ( frac{d y}{d x} )
( left(x^{3}-2 y^{3}right) d x+3 x y^{2} d y=0 )
12
181The 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
182The 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
183Order 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
184If ( 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
185The soultion of ( 3 e^{x} cos ^{2} y d x+(1- )
( left.e^{x}right) cot y d y=0 ) is
12
186Determine 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
187Solve the differential equation:
( (x-sqrt{x y}) d y=y d x )
12
188Solve 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
189Form 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
190If ( 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
191The order of differential equation of all
circles of given radius ( ^{prime} a^{prime} ) is
A . 4
B. 2
( c cdot 1 )
D. 3
12
192Solve ( sec ^{2} x tan y d y+sec ^{2} y tan x d x= )
( mathbf{D} )
12
193What 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
194The 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
195The 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
196Prove 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
197What 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
198An 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
199Solve 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
200Eliminate 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
201The 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
202Solve the differential equation:
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}-boldsymbol{y} tan boldsymbol{x}=mathbf{0} )
12
203Obtain the differential equation from
the relation ( A x^{2}+B y^{2}=1, ) where ( A )
and B are constants.
12
204The 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
205Find 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
206The 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
207If ( 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
208The 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
209The 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
210Form 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
211A 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
212Form 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
213Form 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
214Find 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
215If ( 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
216From 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
217If 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
218Verify 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
219Degree 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
220The 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
221Let ( 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
222Solve 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
223The differential equation for the family of circle
x2 + y2 – 2ay = 0, where a is an arbitrary constant is

(a) (x2 + y²)y’ = 2xy (b) 2(x2 + y2)y’ = xy
(c) (x2 – y2)y’ = 2xy (d) 2(x2 – y2)y’ = xy
12.
12
224Find 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
225If ( 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
227Solve:
( (x+y) frac{d y}{d x}=1 )
12
228The 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
229Find 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
230The 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
231Let ( 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
232A 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
233Which 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
234The 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
236The 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
237A 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
238The 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
239The 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
240The 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
241Solution of the differential equation ydx + (x + xy)dy = 0
is

(a)
log y = Cx
— + log y = C
xy
= C
(c)
—+log y=C
xy
Xy
12
242Assertion
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
243A 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
244If ( 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
245Consider 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
246Form the differential equation
corresponding to ( y=e^{m x} ) by
eliminating ( boldsymbol{m} )
12
247Solution 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
248The 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
249For the following differential equation, find the general solution. ( frac{d y}{d x}+x=1 )12
250The 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
251The 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
253Solve:
( frac{d y}{d x}=x+1 ; ) find ( y ) when ( x=2 )
12
254Find the differential equation of all the
ellipse whose center at origin and axis are along the coordinate axis.
12
255Solution 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
256Show 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
257The 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
258Write 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
259Let 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
260Solution 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
262The 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
263The 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
264Find 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
265Solve 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
266Find the differential equation of the
family of concentric circles ( x^{2}+y^{2}= )
( boldsymbol{a}^{2} )
12
267Form the differential equation of the family of circles in the first quadrant, which touches the coordinate axes.12
268The 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
269Determine 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
270Solve:
(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
271Find 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
272The 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
273The 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
274Order 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
275Solution 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
276Find 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
277A 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
278Assertion
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
2796.
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?
(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
281Form 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
282The 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
283The 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
284Form the differential equation by
eliminating the arbitrary constant ( a ) from the relation ( (x-a)^{2}+y^{2}=1 )
12
285The 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
286Equation 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
287Solve 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
288Solve 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
289Solve:
( boldsymbol{x}^{e}+boldsymbol{e}^{boldsymbol{x}}+boldsymbol{e}^{boldsymbol{e}} )
12
290A 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
291The 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
292Solve:
( 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
294ff ( 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
295The 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
296The 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
297Solve 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
298Order 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
299The 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
300Let 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
301The 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
302Find the age of an object that has been excavated and found to have ( 90 % ) of its
14
A. 378 years old
B. 248 years old
c. 878 years old
D. None of these
12
303Solve ( 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
305Through 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
306Find a differential equation
corresponding to ( y=a x^{2}+b x )
12
307The 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
308dy
It &
= y (log y – log x + 1), then the solution of the
dx
equation is

(a), y log() = ex
(C) log (9) =a
(b) xlog (3) =cy
(a) log (*) –cy
y
12
309Solution 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
310The 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
311Family 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
312The 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
313The 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
314Find the differential equation of family of circles of all of radius ( 5, ) with their
centres on the y-axis.
12
315Solve:
( 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
316Solution 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
317Find the differential equation representing the family of curves ( boldsymbol{y}= )
( a e^{b x+5}, ) where a and ( b ) are arbitrary
constants.
12
318What 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
320Solve the following differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}^{2}+boldsymbol{3} boldsymbol{x}+boldsymbol{7} )12
321Write 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
322Prove 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
323The 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
324Solve the differential equation:
( cos x frac{d y}{d x}+y=sin x )
12
325Form 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
326Let ( 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
329The 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
330The 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
331Solve the following differential ( left(e^{x}+1right) d y=(y+1) e^{x} d x )12
332The 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
333Write the differential equation
representing family of curves ( boldsymbol{y}=boldsymbol{m} boldsymbol{x} )
where ( m ) is arbitrary constant.
12
334The 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
335Form the differential equation
corresponding to ( y=e^{m x} ) by
eliminating m.
12
336Find 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
33719.
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
338The 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
339A 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
340The 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
341ff ( 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
342The 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
343If ( 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
344Solve ( (boldsymbol{x}+boldsymbol{y})^{2} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{a}^{2} )12
345A 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
346D.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
3471.
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
348Find 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
349Obtain a differential equation from the
following equation:
( sin ^{-1} x+sin ^{-1} y=sin ^{-1} c )
12
350Find 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
351The 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
352The 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
353For 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
354The 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
356Solve: ( x^{2} frac{d y}{d x}=frac{y(x+y)}{2} )12
357The 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
358Solve the differential equation:
( left(1+e^{2 x}right) d y+left(1+y^{2}right) e^{x} d x=0 )
12
359The 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
360Which 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
361Solve 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
362The 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
363The 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
364ff ( 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
365Find 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
366If ( 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
367Solve the differential equation: ( frac{d y}{d x}- ) ( x sin ^{2} x=frac{1}{x log x} )12
368Find 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
369Find the deff equation ( left(x^{2}+y^{2}right) d x )
( 2 x y d y=0 )
12
370Find 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
371Solution 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
372The solution of ( frac{d y}{d x}=frac{sqrt{x^{2}-y^{2}}+y}{x} )12
373Determine 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
375Match 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
376Solution 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
377Consider 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?
(a) P=y+x
(b) P=y- x
(c) P+Q=1-x+y+y+692 (d) P-Q=x+y-y-032
12
378Find 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
379Form the differential equation
corresponding to ( y^{2}-2 a y+x^{2}=a^{2} b y )
eliminating a.
12
380If ( 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
381Let ( 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
382In 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
383The 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
385The 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
387Find 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
388Solve 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
38920.
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
390Solve ( : boldsymbol{x}^{2} boldsymbol{y} boldsymbol{d} boldsymbol{x}=left(boldsymbol{x}^{3}+boldsymbol{y}^{3}right) boldsymbol{d} boldsymbol{y}=mathbf{0} )12
391Solution 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
392Solution 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
394Obtain a differential equation by
eliminating ( c ) when it is given
( tan ^{-1} x+tan ^{-1} y=tan ^{-1} c )
12
395Find the differential equation of the family of all striaght lines passing through the origin.12
396If ( 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
397Find 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
398A 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
399Find the order of the differential
equations of all circles in the plane XOY
which have their centres on x-axis and
12
400The 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
401The 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
40211. 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
403Let ( 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
404If ( 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
405Family ( y=a x+a^{3} ) of curve
repersented by the differential equation of degree
( A ). three
B. two
c. one
D. four
12
406The 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
407If 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
408Find 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
409The 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
410If ( 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
411The 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
412Solve the differential equation:
( (x-sin y) d y+tan y d x=0 )
12
413The 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
414The 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
415Solve: ( 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
416Solve ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x} cos boldsymbol{x} )12
417If 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
418Form 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
419If ( 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
420The 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
421The 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
422Solve:
( frac{d y}{d x}=2 x^{2}+x ; ) find ( y ) when ( x=0 )
12
423Find 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
424Represent the following families of curves by forming the corresponding differential equation.(a, b being parameters). ( x^{2}-y^{2}=a^{2} )12
425The 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
426If ( 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
427Form the differential equation
representing the family of curves ( boldsymbol{y}= )
( boldsymbol{m} boldsymbol{x}, ) where ( m ) is arbitrary constant.
12
428Differential 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
429The 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
430Form 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
431What 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
432If ( 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
433Solve
( int frac{d y}{y}=int frac{d x}{x} )
12
434The 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
435Find 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
436The 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
437The 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
438If ( 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
439The 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
440Solve the following differential equation ( cos x frac{d y}{d x}-cos 2 x=cos 3 x )12
441Find ( 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
442The 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
443The 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
444Show that the given differential equation is homogenous and solve them ( boldsymbol{Y}^{prime}=frac{boldsymbol{x}+boldsymbol{y}}{boldsymbol{x}} )12
445The 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
446The number of days so that half the population have flu is
( mathbf{A} cdot 50 )
B. 40
c. 45
D. 25
12
447The 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
448Form 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
449The 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
450A 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
451A 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
452Find 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
453Solve: ( frac{d y}{d x}+x frac{y}{x}=frac{y^{2}}{x^{2}} )12
454Solve 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
456The 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
457If ( 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
458The 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
459JU
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
460Show 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
461Solve: ( frac{d y}{d x}+2 y=sin x )12
462Solve 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
463Form the differential equation from the following primitive, where constant is arbitrary. ( boldsymbol{x} boldsymbol{y}=boldsymbol{a}^{2} )12
464The 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
46614.
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
467Integrating 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
468The 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
469The 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
470The 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
471The 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
472Obtain 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
473Find 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
474The 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
475The solution of ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{y}}{boldsymbol{x}}+sin left(frac{boldsymbol{y}}{boldsymbol{x}}right) ) is12
476The 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
477If 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
478To 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
479If 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
480The 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
481ff ( 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
4824.
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
483The 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
484Consider 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
485The 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
486Form the differential equation from the following primitive, where constant is arbitrary. ( boldsymbol{x} boldsymbol{y}=boldsymbol{a}^{2} )12
487Find 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
488Find 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
48910.
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 
(a) y2 = x2 + 2xy dy
. (b) 2 = x2 – 2xydy
(C) x2 = y2 + xy dy
(d) x2 = y2 + 3xy and
a
12
490If 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
491The 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
492The 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
493If ( 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
494The 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
495Solve:
( (x-y)^{2} frac{d y}{d x}=a^{2} )
12
496If ( (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
497Solve 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
498Family ( 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
499The 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
500Solve the following differential equation
( left(cot ^{-1} y+xright) d y=left(1+y^{2}right) d x )
12
501I.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
502Form the differential equation corresponding to ( y^{2}-2 a y+x^{2}=a^{2} b y )
eliminating a.
12
503The 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
504The differential equation by eliminating ( A, B, C ) from ( y=A e^{2 x}+B e^{3 x}+C e^{2 x} )
is
12
505If 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
506The 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
507A 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
508f ( 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
509Solve ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}+boldsymbol{y} )12
510Write 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
511The 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
512Show that ( y=frac{1}{x} ) is a solution of the differential equation ( frac{d y}{d x}=log x )12
513Let ( 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
514The 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
515Assertion
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
516Particular 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
517Obtain a differential equation by
eliminating ( c ) when it is given
( tan ^{-1} x+tan ^{-1} y=tan ^{-1} c )
12
518Solve the following differential eqauton
( boldsymbol{y}^{2} boldsymbol{d} boldsymbol{y}-boldsymbol{x}^{2} boldsymbol{d} boldsymbol{x}=mathbf{0} )
12
519ff ( 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
52023.
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
521The 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
522Reduce 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
523The 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
524What 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
525The 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
526A 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
527D.E. of family of parabolas symmetrical
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
528Solution 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
529Form 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
530Solve: ( 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
531The 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
532Check 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
533A 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
534A 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
535A 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
536If ( 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
537State 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
538The 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
539If ( 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
540The 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
541I.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
542f ( 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
543Solve the following differential equation
( frac{d y}{d x}+1=sin x )
12
544Find an expression for ( boldsymbol{y} ) given ( frac{d y}{d x}=7 x^{5} )12
545Prove 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
546Solve the following differential equation ( frac{d y}{d x}=1-cos x )12
547For 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
549Solve the differential equation ( y d x+ )
( left(boldsymbol{x}-boldsymbol{y} boldsymbol{e}^{boldsymbol{y}}right) boldsymbol{d} boldsymbol{y}=mathbf{0} )
12
550Order of differential equation of ( y= )
( boldsymbol{m} boldsymbol{x}+boldsymbol{c} )
12
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
552Solve: ( 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
554Form the differential equation
representing the family of curves ( y^{2}= )
( mleft(a^{2}-x^{2}right), ) where ( a ) and ( m ) are
parameters.
12
555Find the degree of each algebraic
expression ( boldsymbol{p} boldsymbol{q}+boldsymbol{p}^{2} boldsymbol{q}-boldsymbol{p}^{2} boldsymbol{q}^{2} )
12
55611. 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?
(2) xy’t V1 – x² = 0
12
557Solve 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
558The 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
559Find 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
560I.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
561The 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
562A 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
563The 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
564Show that the differential equation ( (x-y) frac{d y}{d x}=x+2 y ) is homogeneous12
565The 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
566If ( 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
567Solve ( e^{x} tan y d x+left(1-e^{x}right) sec ^{2} y d y= )
0
12
568What 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
569If ( boldsymbol{x}=tan left(e^{-y}right), ) then show that ( frac{d y}{d x}= )
( frac{e^{-y}}{1+x^{2}} )
12
570Form 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
571The 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
572Solve:
( 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
573The solution of the differential equation ( x d y-y d x=sqrt{x^{2}-y^{2}} d x ) is12
574Solution 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
5752.
The solution of the equation a y
12
– + cxtd
cos
2* + cx? + d.
+ cx
(a) ** + cx+d
+ cx + d
12
576If ( 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
577Find ( frac{d y}{d x} ) when ( y=e^{sqrt{x}+sin x} )12
578Which 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
579Degree 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
580Find the differential equation of the family of all straight lines passing through the origin.12
581Find 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
582Solve ( (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
583The 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
584Form the differential equation corresponding to ( y^{2}-2 a y+x^{2}=a^{2} b y )
eliminating a.
12
585Assertion
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
586The 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
587Solve 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
588Find 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
589The solution of the differential equation
(1+y)+ (x – e tandy – a

(a) re2tanly = etan ‘y+k (b) (x – 2) = ke2 tany
(c) 2xetany = e2 tany +k (d) xetan’ y = tany+k
12
590Write 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
591U
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
592Show 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
593The 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
594The 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
595A 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
596A 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
597Form the different equations of all concentric circles at the origin.12
598Find the general solution of the differential equation ( frac{d y}{d x}-y=cos x )12
599Find 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
600The 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
601The 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
602f ( 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
603By 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
604Form the differential equation from the following primitive where constant is arbitrary. ( boldsymbol{x} boldsymbol{y}=boldsymbol{a}^{2} )12
605The 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
6061.
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
607Form the differential equation of the
family of curves represented by ( y^{2}= )
( (x-c)^{3} )
12
608If 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
609Solve:
( left(x frac{d y}{d x}-yright) tan ^{-1} frac{y}{x}=x )
12
610Differential equation from ( a x^{2}+b y^{2}= )
( mathbf{1} ) is
12
611Find 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
612Solve the following differential equation ( cos x frac{d y}{d x}-cos 2 x=cos 3 x )12
613Form the differential equation
corresponding to ( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x},(boldsymbol{a}, boldsymbol{b}) )
12
614If ( 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
615The 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
616If ( 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
617The 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
618The 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
619A 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
620Write 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
621Obtain 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
622The 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
623The 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
624If ( 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
625If ( 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
626Time 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
627Solve the differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( frac{x^{2}+5 x y+4 y^{2}}{x^{2}} )
12
628The 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
629The 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
631Solve: ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+mathbf{1}=boldsymbol{e}^{boldsymbol{x}+boldsymbol{y}} )12
632The 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
633The 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
634What is the number of arbitrary constants in the particular solution of
differential equation of third order?
A .
в.
( c cdot 2 )
D.
12
6359.
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
wla
12
636Find ( frac{d y}{d x} ) if ( y=sin ^{-1}left(frac{2^{x+1}}{1+4^{x}}right) )12
637Determine 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
638The 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
639Represent the following families of curves by forming the corresponding differential equation.(a, b being parameters). ( x^{2}-y^{2}=a^{2} )12
640The 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
641Find the differential coefficient of
( tan ^{-1} x ) w.r to ( x )
12
642If ( 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
644The 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
645If ( 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
646The 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
647The 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
648Evaluate:
( (1+cos x) d y=(1-cos x) d x )
12
649The 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
650The 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
652At 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
653If 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
654The 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
655Which 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
656The 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
657Obtain 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
658Solve :
( frac{d y}{d x}=x(2 log x+1), operatorname{given} y=0 ) where
( boldsymbol{x}=mathbf{2} )
12
659The 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
660The 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
661Determine 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
662Degree 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
663Which 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
664Consider 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
665The 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
666The 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
667Find 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
668Solve the differential equation ( boldsymbol{x} boldsymbol{d} boldsymbol{y}+ )
( 2 y d x=0 ) when ( x=2, y=1 )
12
669The 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
67016.
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 :

(a) 2ln 18
(b) In 9
(c)
= In 18 (d) In 18

metod
12
671Find 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
672The 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
673If(a + bx) eYx=x, then prove that x
dx
(1983 – 3 Marks)
12
674If ( 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
675If ( 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
676Solve ( : boldsymbol{y} log boldsymbol{y} frac{d boldsymbol{x}}{d boldsymbol{y}}+boldsymbol{x}-log boldsymbol{y}=mathbf{0} )12
677Find the general solution of ( (x+ ) ( left.2 y^{3}right) frac{d y}{d x}=y )12
678What 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
679Consider 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
680Obtain the differential equation of the family of circles touching the y-axis at the origin.12
681The 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
682The 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
683Example 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
684The differential equation representing the family of curves
12 =2c(x+Vc), where c>0, is a parameter, is of order and
degree as follows:

(a) order 1, degree 2 (b) order 1, degree 1
(c) order 1, degree 3 (d) order 2, degree 2
12
685If ( (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
686Find 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
690If 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
691Let ( 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
692Degree 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
693Let ( 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 ) then12
694A 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
695The 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
696If ( 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
697Define particular solution of a differential equation12
698For 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
699Show 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
700Find 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
70110. 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)=
(a) 3
(b) 9
(c) 16
(d) 80
12
702The 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
703If 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
7043.
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
705The 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
706The 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
707The ( 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
708Solve the following differential equation ( frac{d y}{d x}=sec ^{2} x )12
709The 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
711The 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
712Construct 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
713The 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
714The 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
715The 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
716The 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
717The 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
718Distinguish 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
719dy
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
720Solve: ( (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
721The 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
722Assertion
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
723ff ( 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
724The 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
725If ( 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
726Find 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
727Solve: ( sin x frac{d y}{d x}-y=sin x cdot tan frac{x}{2} )12
728Lt 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
729Find 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
730Solve :
( log left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)=boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{y} )
12
731Solve 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
732The 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
733It 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
734Find 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
735Solve the differential equation:
( boldsymbol{y} log boldsymbol{y} boldsymbol{d} boldsymbol{x}-boldsymbol{x} boldsymbol{d} boldsymbol{y}=mathbf{0} )
12
736Obtain 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
737The ( 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
738Solve the following differential equation ( frac{d y}{d x}=e^{x} )12
739Solve: ( frac{d y}{d x}+2 y tan x=sin x )12
740ff ( 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
741Solve the following differential equation ( frac{d y}{d x}=x^{2} )12
74210.
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
743Write 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
744State 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
745The 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
746Order 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
7473.
The degree and order of the differential equation of the
family of all parabolas whose axis is x-axis, are res

(2) 23 (6) 2,1 (c) 1,2 (d) 3,2.
12
748The 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
749The 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
750The 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
751ff ( 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
752Assertion
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
753Find 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
754Write 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
755Solve the following differential equation ( cos x frac{d y}{d x}-cos 2 x=cos 3 x )12
756The 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
757If ( 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
758Degree 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
759The degree of the differential equation ( 1+left(frac{d y}{d x}right)^{2}=x ) is12
760Which 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
761Solve 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
762An 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
763Solve:
( x d y+y d x=x^{2} y d y )
12
764If ( (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
765For 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
766If ( 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
767The 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
768Form the differential equation
representing the family of curves ( boldsymbol{y}= ) ( a sin (x+b), ) where ( a, b ) are arbitrary
constants.
12
769The 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
770The 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
771If ( 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
772If ( 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
773Let ( 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
774The 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
775What 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
776Write 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
777Find 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
778Form 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
779Degree 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

Hope you will like above questions on differential equations and follow us on social network to get more knowledge with us. If you have any question or answer on above differential equations questions, comments us in comment box.