IAS PREVIOUS YEARS QUESTIONS (2017–1983) IAS · 2018-07-31 · ias previous years questions...
Transcript of IAS PREVIOUS YEARS QUESTIONS (2017–1983) IAS · 2018-07-31 · ias previous years questions...
ORDINARY DIFFERENTIAL EQUATIONS / 1IAS PREVIOUS YEARS QUESTIONS (2017–1983)
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
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)
IAS PREVIOUS YEARS QUESTIONS (2017–1983)2 / ORDINARY DIFFERENTIAL EQUATIONS
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
� 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)
� Solve the differential equation
(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)
ORDINARY DIFFERENTIAL EQUATIONS / 3IAS PREVIOUS YEARS QUESTIONS (2017–1983)
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
� 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)
� Using the method of variation of parameters, solve
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
IAS PREVIOUS YEARS QUESTIONS (2017–1983)4 / ORDINARY DIFFERENTIAL EQUATIONS
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
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)
� Solve the differential equation
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)
� Solve the differential equation
3 22
3 2 2
12 2 10 1
d y d y yx x
dx dx x x� �� � � �� �� �
. (15)
� Solve the differential equation
� �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)
ORDINARY DIFFERENTIAL EQUATIONS / 5IAS PREVIOUS YEARS QUESTIONS (2017–1983)
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
� 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)
� Solve the differential equation
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)
� Solve the differential equation
2 4 24 6 secx y xy y x x�� �� � �
by variation of parameters. (15)
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)
IAS PREVIOUS YEARS QUESTIONS (2017–1983)6 / ORDINARY DIFFERENTIAL EQUATIONS
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
� 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)
� Using the method of variation of parameters, solve
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)
1999� Solve the differential equation
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
� � � .
� Solve the differential equation
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
ORDINARY DIFFERENTIAL EQUATIONS / 7IAS PREVIOUS YEARS QUESTIONS (2017–1983)
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
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.
IAS PREVIOUS YEARS QUESTIONS (2017–1983)8 / ORDINARY DIFFERENTIAL EQUATIONS
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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� .
� Solve the differential equation
4 3 2
4 3 2– 2 5 – 8 4 xd y d y d y dy
y edx dx dx dx
� � � .
� Solve the differential equation
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.
� Solve the differential equation
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.
� Solve the differential equation
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
� Solve the equation 2
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
ORDINARY DIFFERENTIAL EQUATIONS / 9IAS PREVIOUS YEARS QUESTIONS (2017–1983)
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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.
� Solve the equation 2
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
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
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)
� Solve the differential equation
22 4
2
d y dyx 2x 4y x
dx dx� � � . (8)
� Solve the differential equation
22dy dy
2 ycot x y .dx dx
� � � � � �� �� �
(15)
� Solve the differential equation
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)
� Using the method of variation of parameters, solve
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)
� Solve the differential equation
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)
2014� Solve the differential equation
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)
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
� Solve the following differential equation
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)
� Solve the differential equation
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)
� Solve the following differential equation
� �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
IFoS PREVIOUS YEARS QUESTIONS (2017–2000)12 / ORDINARY DIFFERENTIAL EQUATIONS
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
� � � �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)
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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
IFoS PREVIOUS YEARS QUESTIONS (2017–2000)14 / ORDINARY DIFFERENTIAL EQUATIONS
HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi60. (M) 9999197625, 01145629987. BRANCH OFFICE(DELHI): 105106, Top Floor, Mukherjee Tower, Mukherjee Nagar,Delhi9. BRANCH OFFICE(HYDERBAD): H.No. 110237, 2nd Floor, Room No. 202 R.K’SKancham’s Blue Sapphire Ashok Nagar Hyd20. (M) 09652351152, 09652661152
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� �� � �
by variation of parameters. (10)
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)