ias previous years questions (2017–1983) ias · 2018-07-31 · ias previous years questions...

ORDINARY DIFFERENTIAL EQUATIONS / 1IAS PREVIOUS YEARS QUESTIONS (2017–1983)

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IASPREVIOUS YEARS QUESTIONS (2017-1983)

SEGMENT-WISE

ORDINARY DIFFERENTIAL EQUATIONS

2017� Find the differential equation representing all the

circles in the x-y plane. (10)� Suppose that the streamlines of the fluid flow are

given by a family of curves xy=c. Find theequipotential lines, that is, the orthogonaltrajectories of the family of curves representing thestreamlines. (10)

� Solve the following simultaneous linear differentialequations: (D+1)y = z+ex and (D+1)z=y+ex where yand z are functions of independent variable x and

.d

Ddx

� (08)

� If the growth rate of the population of bacteria atany time t is proportional to the amount present atthat time and population doubles in one week, thenhow much bacterias can be expected after 4 weeks?

(08)� Consider the differential equation xy p2–(x2+y2–1)

p+xy=0 where .dy

pdx

� Substituting u = x2 and v =

y2 reduce the equation to Clairaut’s form in terms of

u, v and .dv

pdu

� � Hence, or otherwise solve the

equation. (10)� Solve the following initial value differential

equations:20y''+4y'+y=0, y(0)=3.2 and y'(0)=0. (07)

� Solve the differential equation:

23 3 2

24 8 sin( ).

d y dyx x y x x

dx dx� � � (09)

� Solve the following differential equation usingmethod of variation of parameters:

22

22 44 76 48 .

d y dyy x x

dx dx� � � � � (08)

� Solve the following initial value problem usingLaplace transform:

2

29 ( ), (0) 0, (0) 4

d yy r x y y

dx��

where 8sin 0

( ) 0

x if xr x

if x

��

� ��

(17)

2016� Find a particular integral of

2x/2

2

d y x 3y e sin

2dx� � . (10)

� Solve :

� �1tan x

2

dy 1e y

dx 1 x

�

� ��

(10)

� Show that the family of parabolas y2 = 4cx + 4c2 isself-orthogonal. (10)

� Solve :

{y(1 – x tan x) + x2 cos x} dx – xdy = 0 (10)

� Using the method of variation of parameters, solvethe differentail equation

� �2 x dD 2D 1 y e log(x), D

dx� � ��

(15)

� find the genral solution of the eqution

3 22

3 2

d y d y dyx 4x 6 4

dxdx dx� � � . (15)

� Using lapalce transformation, solve the following :

y 2y 8y 0,�� y(0) = 3, y (0) 6� � (10)

2015� Solve the differential equation :

x cosx dy

dx+y (xsinx + cosx) = 1. (10)

https://www.ims4maths.com/coaching/

IAS PREVIOUS YEARS QUESTIONS (2017–1983)2 / ORDINARY DIFFERENTIAL EQUATIONS


� Sove the differential equation :

(2xy4ey + 2xy3 + y) dx + (x2y4ey – x2y2–3x)dy = 0(10)

� Find the constant a so that (x + y)a is the Integratingfactor of (4x2 + 2xy + 6y)dx + (2x2 + 9y + 3x)dy = 0and hence solve the differential eauation. (12)

� (i) Obtain Laplace Inverse transform of

–2 2

11 .

25ss

n es s

��

(ii) Using Laplace transform, solve

y'' + y = t, y(0) = 1, y'(0 = –2. (12)

� Solve the differential equation

x = py – p2 where p = dy

dx

� Solve :

x 4

4

4

d y

dx+ 6 x 3

3

3

d y

dx+ 4 x 2

2

2

d y

dx–2 x

dy

dx–4 y = x 2+ 2

cos(logex).

2014� Justify that a differential equation of the form :

[y + x f(x2 + y2)] dx+[y f(x2 + y2) –x] dy = 0,

where f(x2 +y2) is an arbitrary function of (x2+y2), is

not an exact differential equation and 2 2

1

x y� is

an integrating factor for it. Hence solve thisdifferential equation for f(x2+y2) = (x2+y2)2. (10)

� Find the curve for which the part of the tangentcut-off by the axes is bisected at the point oftangency. (10)

� Solve by the method of variation of parameters :

dy

dx –5y = sin x (10)

� Solve the differential equation : (20)

3 23 2

3 23 8 65cos(log )e

d y d y dyx x x y x

dx dx dx� � � �

� Solve the following differential equation : (15)

22

2 – 2( 1) ( 2) ( – 2) ,xd y dyx x x y x e

dx dx� � � �

when ex is a solution to its correspondinghomogeneous differential equation.

� Find the sufficient condition for the differentialequation M(x, y) dx + N(x, y) dy = 0 to have anintegrating factor as a function of (x+y). What willbe the integrating factor in that case? Hence findthe integrating factor for the differential equation

(x2 + xy) dx + (y2+xy) dy = 0 and solve it. (15)

� Solve the initial value problem

2–2

2 8 sin , (0) 0, '(0) 0td yy e t y y

dt� � � �

by using Laplace-transform

2013� y is a function of x, such that the differential

coefficient dy

dx is equal to cos(x + y) + sin (x + y).

Find out a relation between x and y, which is freefrom any derivative/differential. (10)

� Obtain the equation of the orthogonal trajectory of

the family of curves represented by rn = a sin n� ,

(r, � ) being the plane polar coordinates. (10)


(5x3 + 12x2 + 6y2) dx + 6xydy = 0. (10)

� Using the method of variation of parameters, solve

the differential equation 2

2

d y

dx+ a2y = sec ax. (10)

� Find the general solution of the equation (15)

x2 2

2

d y

dx+ x

dy

dx + y = ln x sin (ln x).

� By using Laplace transform method, solve thedifferential equation (D2 + n2) x = a sin (nt + �),

D2 =2

2

d

dtsubject to the initial conditions x = 0 and

dx

dt = 0, at t = 0, in which a, n and � are constants.

(15)

2012

� Solve

2

2 2

( )

2 ( ) 2 ( )

2

(1 ) 2

x y

x y x y

dy xy e

dx y e x e�

� � (12)

� Find the orthogonal trajectories of the family of

curves 2 2 .x y ax� � (12)




� Using Laplace transforms, solve the intial value

problem 2 , (0) 1, (0) 1�� ty y y e y y

(12)

� Show that the differential equation

� �2 2 2(2 log ) 1 0� � � �xy y dx x y y dy

is not exact. Find an integrating factor and hence,

the solution of the equation. (20)

� Find the general solution of the equation

212 6 .�� y y x x (20)

� Solve the ordinary differential equation

2( 1) (2 1) 2 (2 3)x x y x y y x x�� (20)

2011� Obtain the soluton of the ordinary differential

equation 2(4 1) ,

dyx y

dx� � � if y(0) = 1. (10)

� Determine the orthogonal trajectory of a family of

curves represented by the polar equation

r = a(1 – cosθ), (r, θ) being the plane polar

coordinates of any point. (10)

� Obtain Clairaut’s orm of the differential equation

2 .dy dy dy

x y y y adx dx dx

� ��

Also find its

general solution. (15)

� Obtain the general solution of the second order

ordinary differential equation

2 2 cos ,xy y y x e x�� where dashes denote

derivatives w.r. to x. (15)


the second order differedifferential equation

2

24 tan 2 .

d yy x

dx� � (15)

� Use Laplace transform method to solve the

following initial value problem:

2

20

2 , (0) 2 and 1t

t

d x dx dxx e x

dt dt dt �

� � � � � � (15)

2010� Consider the differential equation

, 0y x x��

where � is a constant. Show that–

(i) if �(x) is any solution and �(x) = �(x) e–�x,

then �(x) is a constant;

(ii) if ��< 0, then every solution tends to zero asx �� . (12)

� Show that the diffrential equation

2 2(3 – ) 2 ( 3 ) 0y x y y x y��

admits an integrating factor which is a function of(x+y2). Hence solve the equation. (12)

� Verify that

� � � �1 1log ( ) ( ) (log ( ))

2 2xye eMx Ny d xy Mx Ny d� � �

= M dx + N dy

Hence show that–

(i) if the differential equation M dx + N dy = 0 ishomogeneous, then (Mx + Ny) is an integrating

factor unless 0;Mx Ny� �

(ii) if the differential equation

0Mdx Ndy� � is not exact but is of the form

1 2( ) ( ) 0f xy y dx f xy x dy� �

then (Mx – Ny)–1 is an integrating factor unless

0Mx Ny� � . (20)

� Show that the set of solutions of the homogeneous

linear differential equation

( ) 0y p x y� � �

on an interval [ , ]I a b� forms a vector subspace

W of the real vector space of continous functionson I. what is the dimension of W?. (20)

� Use the method of undetermined coefficiens to find

the particular solution of 2sin (1 ) xy y x x e��

and hence find its general solution. (20)

2009� Find the Wronskian of the set of functions

� �3 33 ,| 3 |x x




on the interval [–1, 1] and determine whether theset is linearly dependent on [–1, 1]. (12)

� Find the differential equation of the family of circles

in the xy-plane passing through (–1, 1) and (1, 1).(20)

� Fidn the inverse Laplace transform of

1( ) ln

5

sF s

s

�� . (20)

� Solve: 2

2 2 3

( ), (0) 1

3 – – 4

dy y x yy

dx xy x y y

�� . (20)

2008� Solve the differential equation

� �3 2 0ydx x x y dy� � � . (12)

� Use the method of variation of parameters to find

the general solution of 2 4– 4 6 – sinx y xy y x x�� .

(12)

� Using Laplace transform, solve the initial value

problem 3– 3 2 4 ty y y t e�� with y(0) = 1,

(0) 1y� � � . (15)


3 2– 3 sin(ln ) 1x y x y xy x�� . (15)

� Solve the equation 22 0y xp yp� � � where dy

pdx

� .

(15)

2007� Solve the ordinary differential equation

cos3dy

xdx

21–3 sin3 sin6 sin 3

2y x x x� � , 0

2x

�� .

(12)

� Find the solution of the equation

2 –4dy

xy dx xdxy� � . (12)

� Determine the general and singular solutions of the

equation

12–2

1dy dy dy

y x adx dx dx

� ��

‘a’ being a

constant. (15)

� Obtain the general solution of 3 2– 6 12 –8D D D� ��

2 –912

4x xy e e

� ��

, where d

Ddx

� . (15)

� Solve the equation 2

2 3

22 3 – 3

d y dyx x y x

dx dx� � . (15)

� Use the method of variation of parameters to findthe general solution of the equation

2

23 2 2 xd y dy

y edx dx

� � � . (15)

2006� Find the family of curves whose tangents form an

angle 4� with the hyperbolas xy=c, c > 0. (12)

� Solve the differential eqaution

� �–132 2– 0xxy e dx x y dy� � . (12)

� Solve � � � �–12 – tan1 – 0y dyy x e

dx� � � . (15)

� Solve the equation x2p2 + yp (2x + y) + y2 =0 usingthe substituion y = u and xy=v and find its singular

solution, where dy

pdx

� . (15)


3 22

3 2 2

12 2 10 1

d y d y yx x

dx dx x x� ��

. (15)


� �2 – 2 2 tanxD D y e x� � , where d

Ddx

� ,

by the method of variation of parameters. (15)

2005� Find the orthogonal trejectory of a system of co-

axial circles x2+y2+2gx+c=0, where g is the parameter. (12)

� Solve 2 2 2 2 1

dyxy x y x y

dx� � � � . ( 12)

� Solve the differential equation (x+1)4 D3+2 (x+1)3

D2–(x+1)2 D +(x+1)y = 1

1x �. (15)




� Solve the differential equation (x2+y2) (1+p)2 –2

(x+y) (1+p) (x+yp) + (x+yp)2 = 0, where dy

pdx

� , by

reducing it to Clairaut’s form by using suitablesubstitution. (15)

� Solve the differential equation (sin x–x cos x)

– sin sin 0y x xy y x�� given that siny x� is a

solution of this equation. (15)


2 – 2 2 log , 0x y xy y x x x��

by variation of parameters. (15)

2004� Find the solution of the following differential

equation 1

cos sin 22

dyy x x

dx� � . (12)

� Solve y(xy+2x2y2) dx + x(xy –x2y2) dy = 0. (12)

� Solve � �4 24 5 ( cos )xD D y e x x� � � � . (15)

� Reduce the equation (px–y) (py+x) = 2p where

dyp

dx� to Clairaut’s equation and hence solve it.

(15)

� Solve (x+2) 2

2– (2 5) 2 ( 1) xd y dy

x y x edx dx

� � � � . (15)

� Solve the following differential equation

22 2

2(1 ) 4 (1 )

d y dyx x x y x

dx dx� � � � � . (15)

2003� Show that the orthogonal trajectory of a system of

confocal ellipses is self orthogonal. (12)

� Solve log .xdyx y y xye

dx� � (12)

� Solve (D5–D) y = 4 (ex+cos x + x3), where d

Ddx

� .

(15)

� Solve the differential equation (px2 + y2) (px + y) =

(p + 1)2 where dy

pdx

� , by reducing it to Clairaut’ss

form using suitable substitutions. (15)

� Solve � � � � � �21 1 sin 2 log 1 .x y x y y x� � � � � � � ��

(15)


2 4 24 6 secx y xy y x x��


2002

� Solve 3 23dy

x y x ydx

� � . (12)

� Find the values of �� for which all solutions of

22

23 – 0

d y dyx x y

dx dx�� tend to zero as .x�� (12)

� Find the value of constant � such that the following

differential equation becomes exact.

� � � �2 22 3 3 0y ydyxe y x e

dx��

Further, for this value of �, solve the equation.(15)

� Solve 4

6

dy x y

dx x y

� ��

� �. (15)

� Using the method of variation of parameters, find

the solution of 2

22 sinxd y dy

y xe xdx dx

� � � with

y(0) = 0 and 0

0x

dy

dx �

� � ��

. (15)

� Solve (D–1) (D2–2 D+2) y = ex where dD

dx� . (15)

2001� A continuous function y(t) satisfies the differential

equation

1

2

1 ,0 1

2 2 3 ,1 5

te tdy

dt t t t

��

If y(0) = –e, find y(2). (12)

� Solve 2

2

2

d yx

dx

23 loge

dyx y x x

dx� � � . (12)




� Solve 2

2

(log )log e

e

y ydy yy

dx x x� � . (15)

� Find the general solution of ayp2+(2x–b) p–y=0,

a>o. (15)

� Solve (D2+1)2 y = 24x cos x

given that y=Dy=D2y=0 and D3y = 12 when x = 0.

(15)


2

24 4 tan 2

d yy x

dx� � . (15)

2000

� Show that 2

23 4 – 8 0

d y dyx y

dx dx� � has an integral

which is a polynomial in x. Deduce the generalsolution. (12)

� Reduce2

2Q

d y dyP y R

dx dx� � � , where P, Q, R are

functions of x, to the normal form.

Hence solve2

22

2– 4 (4 –1) –3 sin2 .xd y dy

x x y e xdx dx

� �

(15)

� Solve the differential equation y = x–2a p+ap2. Fnd

the singular solution and interpret it geometrically.(15)

� Show that (4x+3y+1)dx+(3x+2y+1) dy = 0 represents

a family of hyperbolas with a common axis andtangent at the vertex. (15)

� Solve – ( –1)dy

x y xdx

� 2

21

d yx

dx

� ��

� � by the

method of variation of parameters. (15)


122 2

2 2

1xdx ydy x y

xdy ydx x y

� ��

� Solve 3 2

3 2– 3 4 – 2 cos .xd y d y dy

y e xdx dx dx

� � �

� By the method of variation of parameters solve the

differential equation 2

2

2sec( )

d ya y ax

dx� � .

1998

� Solve the differential equation 23 xdy

xy y edx

��

� Show that the equation (4x+3y+1) dx + (3x+2y+1)

dy = 0 represents a family of hyperbolas having asasymptotes the lines x+y = 0; 2x+y+1=0. (1992)

� Solve the differential equation y = 3px + 4p2.

� Solve 2

4 2

2– 5 6 ( 9)xd y dy

y e xdx dx

� � � .


2

22 sin .

d y dyy x x

dx dx� � �

1997

� Solve the initial value problem 2 3

dy x

dx x y y�

�,

y(0)=0.

� Solve (x2–y2+3x–y) dx + (x2–y2+x–3y)dy = 0.

� Solve 4 3 2

24 36 11 6 20 sinxd y d y d y dy

e xdx dxdx dx

��

� Make use of the transformation y(x) = u(x) sec x to

obtain the solution of – 2 tan 5 0y y x y�� ;

y(0)=0; (0) 6y� � .

� Solve (1+2x)2 2

2– 6

d y

dx (1+2x) 16

dyy

dx� = 8 (1+2x)2;

(0) 0 and (0) 2y y�� .

1996

� Solve x2 (y–px) = yp2;dy

pdx

� ��

.

� Solve y sin 2x dx – (1+y2 + cos2x) dy = 0.

� Solve 2

2

d y

dx+ 2

dy

dx+10y+37 sin 3x = 0. Find the value

of y when 2x �� , if it is given that y=3 and




0dy

dx� when x=0.

� Solve 4 3 2

4 3 22 – 3

d y d y d y

dx dx dx� = x2 + 3e2x + 4 sin x.

� Solve 3 2

3 2

3 23 log .

d y d y dyx x x y x x

dx dx dx� � � � �

1995� Solve (2x2+3y2–7)xdx– (3x2+2y2–8) y dy = 0.

� Test whether the equation (x+y)2 dx – (y2–2xy–x2)

dy = 0 is exact and hence solve it.

� Solve 3

3

3

d yx

dx+

22

22

d yx

dx+2y =10

1x

x� ��

. (1998)

� Determine all real valued solutions of the equation

– 0y iy y iy�� , dy

ydx

� � .

� Find the solution of the equation 4 8cos2y y x��

given that 0y � and 2y� � when x = 0.

1994

� Solve dy

dx+ x sin 2y = x3cos2y..

� Show that if 1 Q

–Q

P

y x

� ��

is a function of x only

say, f(x), then ( )( )

f x dxF x e�� is an integrating factor

of Pdx + Qdy = 0.

� Find the family of curves whose tangent form angle

4� with the hyperbola xy = c.

� Transform the differential equation

23 5

2cos sin 2 cos 2cos

d y dyx x y x x

dx dx� � � into one

having z an independent variable where z = sin xand solve it.

� If 2

2

d x

dt+ ( – ) 0

gx a

b� , (a, b and g being positive constants)

and x = a’ and 0dx

dt� when t=0, show that

� �cosg

x a a a tb

�� .

� Solve (D2–4D+4) y = 8x2e2x sin 2x, where, d

Ddx

� .

1993

� Show that the system of confocal conics

2 2

2 21

x y

a b� ��

� � is self orthogonal.

� Solve 1

1 cosy y dxx

� ��

+ � �log – sinx x x y�

dy = 0.

� Solve 2

202

d yw y

dx� = a coswt and discuss the nature

of solution as 0.w w�

� Solve (D4+D2+1) y = 2– 3cos

2

x xe

� ��

.

1992

� By eliminating the constants a, b obtain the

differential equation of which xy = aex +be–x +x2 is a

solution.

� Find the orthogonal trajectories of the family of

semicubical parabolas ay2=x3, where a is a variable

parameter.

� Show that (4x+3y+1) dx + (3x+2y+1) dy = 0

represents hyperbolas having the following lines

as asymptotes

x+y = 0, 2x+y+1 = 0. (1998)

� Solve the following differential equation y (1+xy)

dx+x (1–xy) dy = 0.

� Solve the following differential equation (D2+4) y =

sin 2x given that when x = 0 then y = 0 and 2dy

dx� .

� Solve (D3–1)y = xex + cos2x.

� Solve (x2D2+xD–4) y = x2.




1991� If the equation Mdx + Ndy = 0 is of the form f

1 (xy).

ydx + f2 (xy). x dy = 0, then show that

1

–Mx Ny is

an integrating factor provided Mx–Ny� 0.

� Solve the differential equation.

(x2–2x+2y2) dx + 2xy dy = 0.

� Given that the differential equation (2x2y2 +y) dx –(x3y–3x) dy = 0 has an integrating factor of the formxh yk, find its general solution.

� Solve 4

4

4– sin

d ym y mx

dx� .


4 3 2

4 3 2– 2 5 – 8 4 xd y d y d y dy

y edx dx dx dx

� � � .


2–

2– 4 – 5 xd y dy

y xedx dx

� , given that y = 0 and

0dy

dx� , when x = 0.

1990

� If the equation �n+a1�n–1+...........+ na = 0 (in

unknown �) has distinct roots �1,�

2, ............. n� .

Show that the constant coefficients of differentialequation

1

1 1

n n

n n

d y d ya

dx dx

�

�� +..............+ an–1

n

dya b

dx� � has the

most general solution of the form

y = c0(x) + c

1e�1x + c

2e�2x + ............... + c

ne�nx .

where c1, c

2 ....... c

n are parameters. what is c

0(x)?

� Analyse the situation where the ��– equation in (a)

has repeated roots.


22

22 0

d y dyx x y

dx dx� � � is explicit form. If your

answer contains imaginary quantities, recast it in aform free of those.

� Show that if the function 1

( )t f t� can be integrated

(w.r.t ‘t’), then one can solve ( )yx

dyf

dx� , for any

given f. Hence or otherwise.

3 20

3 6

dy x y

dx x y

� ��

� �

� Verify that y = (sin–1x)2 is a solution of (1–x2) 2

2

d y

dx

2dy

xdx

� � . Find also the most general solution.

1989� Find the value of y which satisfies the equation

(xy3–y3–x2ex) + 3xy2 dy

dx=0; given that y=1 when x =

1.

� Prove that the differential equation of all parabolas

lying in a plane is

23–2

2

d d y

dx dx

� ��

= 0.


3 22

3 2– 6 1

d y d y dyx

dx dx dx� � � .

1988

� Solve the differential equation 2

2

d y

dx –2

dy

dx= 2ex

sin x.

� Show that the equation (12x+7y+1) dx + (7x+4y+1)dy = 0 represents a family of curves having asasymptotes the lines 3x+2y–1=0, 2x+y+1=0.

� Obtain the differential equation of all circles in a

plane n the form 3

3

d y

dx

22 2

21 –3 0

dy dy d y

dx dx dx

� � � ��

.

1987


2

d yx

dx+ (1–x)

dy

dx = y + ex

� If f(t) = tp–1, g(t) = tq–1 for t > 0 but f(t) = g(t) = 0

for 0t � , and h(t) = f * g, the convolution of f, g

show that 1( ) ( )( ) ; 0

( )p qp q

h t t tp q

� ��

� � and p, q are




positive constants. Hence deduce the formula

( ) ( )( , )

( )

p qB p q

p q

� ��

� �.

1985

� Consider the equation y� +5y = 2. Find that solution

� of the equation which satisfies ��(1) = 3�� (0).

� Use Laplace transform to solve the differential

equation 2 , 't dx x x e

dt� ��

such that

(0) 2, (0) 1x x�� .

� For two functions f, g both absolutely integrable

on ( , )�� , define the convolution f * g.

If L(f), L(g) are the Laplace transforms of f, g showthat L (f * g) = L(f). L(g).

� Find the Laplace transform of the function

1 2 (2 1)( )

–1 (2n+1) t (2n+2)

n t nf t

� ��

� � ��

n = 0, 1, 2, ..............

1984

� Solve 2

2

d y

dx + y = sec x.

� Using the transformation k

uy

x� , solve the

equation x y��+ (1+2k) y� +xy = 0.

� Solve the equation 2( 1) cos 2D x t t� � ,

given that 0 1 0x x� � by the method of Laplace

transform.

1983

� Solve 2

2( 1) –d y dy

x x y xdx dx

� � � .

� Solve (y2+yz) dx+(xz+z2) dy + (y2–xy) dz = 0.


22

d y dyy t

dt dt� � � by the

method of Laplace transform, given that y = –3 whent = 0, y = –1 when t = 1.


IFoS PREVIOUS YEARS QUESTIONS (2017–2000)10 / ORDINARY DIFFERENTIAL EQUATIONS


IFoSPREVIOUS YEARS QUESTIONS (2017-2000)

SEGMENT-WISEORDINARY DIFFERENTIAL EQUATIONS

(ACCORDING TO THE NEW SYLLABUS PATTERN) PAPER II

2017

� Solve (2D3 – 7D2 + 7D – 2) y = e–8x where d

D .dx

�

(8)


22 4

2

d y dyx 2x 4y x

dx dx� � � . (8)


22dy dy

2 ycot x y .dx dx

� � � � � ��

(15)


33x 2ydy dy

e 1 e 0.dx dx

� � � ��

(10)

� Solve 2

2

d y4y tan 2x

dx� � by using the method of

variation of parameter. (10)

2016� Obtain the curve which passes through (1, 2) and

has a slope 2

2xy

x 1

��

�. Obtain one asymptote to

the curve. (8)

� Solve the dE to get the particular integral of

4 22

4 2

d y d y2 y x cos x

dx dx� � � . (8)


22 2 x

2

d y dyx x y x e

dx dx� � � . (10)

� Obtain the singular solution of the differentialequation

� �2 2 2 2 dyy 2pxy p x 1 m , p

dx� � � � � . (10)

� Solve the differential equation (10)

� �2dyy y sin x cos x

dx� � � .

2015� Reduce the differential equation

x2p2+yp(2x+y)+y2=0, dy

pdx

� to Clairaut’s form.

Hence, find the singular solution of the equation.(8)


22

2 2

13

(1 )

d y dyx x y

dx dx x� � �

� (8)

� Solve 2

3 3 22 4 8 sin

d y dyx x y x x

dx dx� � � by changing

the independent variable. (10)

� Solve 4 2 /2 3( 1) cos ,

2x x

D D y e� � ��

� �

where .d

Ddx

� (10)


22 ,dy

y px p y pdx

� � �

and obtain the non-singular solution (8)� Solve

44

4 16 sin .d y

y x xdx

� � � (8)


ORDINARY DIFFERENTIAL EQUATIONS / 11IFoS PREVIOUS YEARS QUESTIONS (2017–2000)



3

2

2tan .

dy y x yx

dx x y x� � � (10)

� Solve by the method of variation of parameters

3 2 cos .y y y x x�� (10)

� Solve the D.E.

3 2

3 23 4 2 cos .xd y d y dyy e x

dx dx dx� � � � � (10)

2013� Solve

3 2sin 2 cosdy

x y x ydx

� � (8)


22

22 4 (4 1) 3 sin 2xd y dy

x x y e xdx dx

� � � � �

by changing the dependent variable. (13)

� Solve

3 23

( 1) sin2

x

D y e x� �

� � � ��

where .d

Ddx

� (13)

� Apply the method of variation of parameters tosolve

21

2 2(1 )xd yy e

dx�� (13)

2012

� Solve tan

(1 ) sec .1

xdy yx e y

dx x� � �

� (8)

� Solve and find the singular solution of

3 2 2 3 0x p x py a� � � (8)

� Solve: 22

2

20

d y dyx y x y

dx dx� ��

(10)

� Solve 4 2

2

4 22 cos .

d y d yy x x

dx dx� � � (10)

� Solve 2

dy dyx y

dx dx� ��

(10)

� Solve 2

2 2

23 (1 )

d y dyx x y x

dx dx�� (10)

2011� Find the family of curves whose tangents form an

angle / 4� With hyperbolas .xy c� (10)

� Solve 2

2 2 tan 5 sec . .xd y dyx y x e

dxdx� � � (10)

� Solve 2 22 cotp py x y� � Where dy

pdx

� . (10)

�� Solve

� � � �24 4 3 3 2 26 9 3 1 1 log ,x D x D x D xD y x� � � � � �

Where .d

Ddx

� (15)

�� Solve 4 2 2( 1) sin 2 ,xD D y ax be x�� where

dD

dx� (15)

2010

� Show that � �cos x y� is an integrating factor of

� �tan 0.y dx y x y dy� ��

Hence solve it (8)

� Solve 2

2 2 sinxd y dyy xe x

dxdx� � � (8)


� �2sin 6dy

x ydx

� � � (8)

� Find the general solution of

� �2

22 2 1 0

d y dyx x y

dxdx� � � � (12)

� Solve

22 2

21 1 xd d

y x edx dx

� ��

(10)

� Solve by the method of variation of parameters thefollowing eqaution




� � � �2 22 2

21 2 2 1d y dy

x x y xdxdx

� � � � � (10)

2009

� Solve 2 3sec 2 tan

dyy x y x

dx� � (10)

� Find the 2nd order ODE for which xe and 2 xx e

are solutions. (10)

� Solve � � � �3 2 2 32 2 0y yx dx xy x dy� � � � . (10)

� Solve

2

2 cos 1 0.dy dy

hxdx dx

� � � � ��

(8)

� Solve 3 2

23 23 3 xd y d y dy

y x edxdx dx

�� (10)

� Show that 2xe is a solution of

� �2

22 4 4 2 0

d y dyx x y

dxdx� � � � . (12)

Find a second independent solution.

2008

� Show that the functions � � 21y x x and�

� � 22 logey x x x� are linearly independent obtain

the differential equation that has � � � �1 2y x and y x

as the independent solutions. (10)

� Solve the following ordinary differential equationof the second degree :

� �2

2 3 0dy dy

y x ydx dx

� � � � � ��

(10)

� Reduce the equation 2dy dy dy

x y x ydx dx dx

� ��

to clairaut’s form and obtain there by the singularintegral of the above equation. (10)

� Solve

� � � � � �2

2

21 1 4coslog 1e

d y dyx x y x

dxdx� � � � � � (10)

� Find the general solution of the equation

� �2

2 cot 1 cot sin .xd y dyx x y e x

dxdx� � � � (10)

2007� Find the orthogonal trajectories of the family of the

curves 2 2

2 2 1,x y

a b�

��

� being a parameter.. (10)

� Show that 2xe and 3xe are linearly independent

solutions of 2

2 5 6 0.d y dy

ydxdx

� � � Find the general

solution when � �0

0 0 1dy

y anddx

�� (10)

� Find the family of curves whose tangents form an

angle 4� � with the hyperbola .xy c� (10)

� Apply the method of variation of parametes to

solve � �2 2 cosD a y ec ax� � (10)

� Find the general solution of � �2

22

1 2d y

x xdx

� �

3 0dy

ydx

� � solution of it. (10)

2006

� From 2 2 2 2 0,x y ax by c� � � � � derive differential

equation not containing, , .a b or c (10)

� Discuss the solution of the differential equation

22 21

dyy a

dx

� ��

(10)

� Solve � �2

2 1 xd y dyx x y e

dxdx� � � � (10)

� Solve 4

4sin

d yy x x

dx� � (10)

� Solve 2

2 22

xd y dyx x y x e

dxdx� � � (10/2008)

� Reduce

� �2

2 2 1 0dy dy

xy x y xydx dx

� � � � � � ��


ORDINARY DIFFERENTIAL EQUATIONS / 13IFoS PREVIOUS YEARS QUESTIONS (2017–2000)


to clairaut’s form and find its singular solution.(10)

2005� Form the differential equation that represents all

parabolas each of which has latus rectum 4a and

Whose are parallel to the x-axis. (10)

� (i) The auxiliary polynomial of a certain

homogenous linear differential equation with

constant coefficients in factored form is

� � � �64 2P m m m� � � �32 6 25m m� � .

What is the order of the differential equationand write a general solution ?

(ii) Find the equation of the one-parameter family

of parabolas given by 2 22 ,y cx c� � C real

and show that this family is self-orthogonal.(10)

� Solve and examine for singular solution the

following equation � �2 2 2 2 22 0P x a pxy y b� � � � �

(10)

� Solve the differential equation 2

2 9 sec3d y

y xdx

� �

(10)

� Given 1

y xx

� � is one solution solve the

differential equation 2

22

0d y dy

x x ydxdx

� � �

reduction of order method. (10)

� Find the general solution of the defferential

equation 2

2 2 3 2 10sinxd y dyy y e x

dxdx� � � � by the

method of undertermined coefficients. (10)

2004� Dertermine the family of orthgonal trajectories of

the family xy x ce�� (10)

� Show that the solution curve satisfying

� �2x xy� 3'y y� Where 1 ,y as x� �� is a

conic section. Indentify the curve. (10)

� Solve � � � � � �� 21 " 1 ' 4cos ln 1x y x y y x� � � � � �

� � � �, 0 1, 1 cos1.y y e� � � (10)

� Obtain the general solution of

2" 2 ' 2 4 xy y y e x�� sin .x (10)

� Find the general solution of � �3 2xy y dx� �

� �2 2 4 0x y x y dy� � � (10)

� Obtain the general solution of � �4 3 22 2D D D D� � �

2 ,xy x e� � Where y

dyD

dx� . (10)

2003� Find the orthogonal trajectories of the family of co-

axial circles 2 2 2 0x y gx c� � � � Where g is a

parameter. (10)

� Find the three solutions of 3 2

3 22 2 0

d y d y dyy

dxdx dx� � � �

Which are linealy independent on every realinterval. (10)

� Solve and examine for singular solution:

� �2 2 2 22 1 .y pxy p x m� � � � (10)

� Solve 3 2

3 23 2

12 2 10

d y d yx x y x

xdx dx� ��

(10)

� Given y x� is one solutions of

� �2

321 2 2 0

d y dyx x y

dxdx� � � � find another linearly

independent solution by reducing order and writethe general solution. (10)

� Solve by the method of variation of parameters

22

2 sec ,d y

a y axdx

� � a real. (10)

2002

� If � �4 nxD a e� is denoted by z , prove that

2 3

2 3, ,

z z zz

n n n

� � ��

all vanish when n a� . Hence




show that 2, , ,nx nx nxe xe x e x nx� are all solutions of

� �40D a y� � . Here D Stands for .

d

dx (10)

� Solve � �224 3 1 0xp x� � � and examine for singular

solutions and extraneous loci. Interpret the resultsgeometrically. (10)

� (i) Form the differential equation whose primitive is

cos sinsin cos

x xy A x B x

x x� � � ��

(ii) Prove that the orthogonal trajectory of systemof parabolas belongs to the system itself. (10)

� Using variation of parameters solve the differentialequation

� � 22

22 4 4 1 3 sin 2 .xd y dy

x x y e xdxdx

� � � � � (10)

� (i) Solve the equation by finding an integratingfactor

of � �2 sin cos 0.x ydx x ydy� � �

(ii) Verify that � � 2x x� � is a solution of

2

2" 0y y

x� � and find a second independent

solution. (10)

� Show that the solution of � �2 1 1 0,nD y� � �

consists of xAe and n paris of terms of the form

� �cos sin ,axr re b x c x� �� Where

2cos

2 1

ra

n

��

�

and 2

sin2 1

r

n

��

�, 1,2.....,r n� and ,r rb c are

arbitrary constants. (10)

2001� A constant coefficient differential equation has

auxiliary eqution expressible in factored form as

� � � � � �223 21 2 5 .P m m m m m� � � � What is the

order of the differential equation and find its generalsolution. (10)

� Solve � �2

2 22 0dy dy

x y x y ydx dx

� � � � � ��

(10)

� Using differential equations show that the system

of confocal conics given by 2 2

2 2 1,x y

a b� ��

� �is self othogonal. (10)

� Solve � �2

2 22

1 0d y dy

x x a ydxdx

� � � � given that

1sina xy e�

� is one solution of this equation. (10)

� Find a general solution tan ,ny y x� �2 2x� ��


2000

� Solve � �� 2 22 2 1 2 1 0x y P x y p x yp x yp� � � � � � � � �

.dy

Pdx

� Interpret geometrically the factors in the P-

and C-discriminants of the equation � �3 28 12 9p x y p� �

(20)� Solve

� �2 2

2 4

20

d y dy ai y

x dxdx x� � �

� � � �2

2 42 tan 3cos 2 cos cos .

d y dyii x x y x x

dxdx� � � �

by varying parameters. (20/2007)


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