Ordinary Differential Equations / 1 Ias Previous Years Questions (2017-1983) Segment-Wise Ordinary Differential Equations

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

d2 y 2017 � 2 �9y � r ( x ), y (0) � 0, y (0) � 4 � Find the differential equation representing all the dx circles in the x-y plane. (10) �8sinx if 0 � x � � � Suppose that the streamlines of the fluid flow are where r() x � � (17) given by a family of curves xy=c. Find the � 0 if x � � equipotential lines, that is, the orthogonal trajectories of the family of curves representing the streamlines. (10) 2016 � Solve the following simultaneous linear differential � Find a particular integral of equations: (D+1)y = z+ex and (D+1)z=y+ex where y and z are functions of independent variable x and d2 y x 3 �y � ex/2 sin . (10) d dx 2 2 D � . (08) dx � Solve : � If the growth rate of the population of bacteria at dy 1 �1 any time t is proportional to the amount present at � �etan x � y � (10) that time and population doubles in one week, then dx 1� x 2 how much bacterias can be expected after 4 weeks? � Show that the family of parabolas y2 = 4cx + 4c2 is (08) self-orthogonal. (10) � Consider the differential equation xy p2–(x2+y2–1) � Solve : dy p+xy=0 where p � . Substituting u = x2 and v = {y(1 – x tan x) + x2 cos x} dx – xdy = 0 (10) dx � Using the method of variation of parameters, solve y2 reduce the equation to Clairaut’s form in terms of the differentail equation dv � u, v and p � . Hence, or otherwise solve the d du �2 � �x � � D� 2D � 1y � e log(x),� D � � (15) equation. (10) �dx � � Solve the following initial value differential � find the genral solution of the eqution equations: 20y''+4y'+y=0, y(0)=3.2 and y'(0)=0. (07) d3 y d2 y dy x2 � 4x � 6 � 4 . (15) � Solve the differential equation: dx3 dx2 dx

2 d y dy 3 3 2 � Using lapalce transformation, solve the following : x 2 � �4x y � 8 x sin( x ). (09) dx dx y�� 2y � � 8y � 0, y(0) = 3, y� (0)� 6 (10) � Solve the following differential equation using method of variation of parameters: 2015 2 d y dy 2 � �2y � 44 � 76 x � 48 x . (08) � Solve the differential equation : dx2 dx � Solve the following initial value problem using dy x cosx +y (xsinx + cosx) = 1. (10) Laplace transform: dx

� Sove the differential equation : � Find the sufficient condition for the differential (2xy4ey + 2xy3 + y) dx + (x2y4ey – x2y2–3x)dy = 0 equation M(x, y) dx + N(x, y) dy = 0 to have an (10) integrating factor as a function of (x+y). What will be the integrating factor in that case? Hence find � Find the constant a so that (x + y)a is the Integrating the integrating factor for the differential equation factor of (4x2 + 2xy + 6y)dx + (2x2 + 9y + 3x)dy = 0 (x2 + xy) dx + (y2+xy) dy = 0 and solve it. (15) and hence solve the differential eauation. (12) � Solve the initial value problem � (i) Obtain Laplace Inverse transform of 2 d y –2t � �1 � s –� s � 2 �y � 8 e sin t , y (0)� 0, y '(0) � 0 ��n�1�2 � � 2 e �. dt � �s � s � 25 � by using Laplace-transform (ii) Using Laplace transform, solve y'' + y = t, y(0) = 1, y'(0 = –2. (12) 2013 � Solve the differential equation � y is a function of x, such that the differential dy x = py – p2 where p = dy dx coefficient is equal to cos(x + y) + sin (x + y). dx � Solve : Find out a relation between x and y, which is free from any derivative/differential. (10) d4 y d3 y d2 y dy x 4 +6 x 3 +4 x 2 –2 x –4 y =x 2+ 2 dx4 dx3 dx2 dx � Obtain the equation of the orthogonal trajectory of the family of curves represented by rn = a sin n� , cos(logex). (r, � ) being the plane polar coordinates. (10) 2014 � Solve the differential equation (5x3 + 12x2 + 6y2) dx + 6xydy = 0. (10) � Justify that a differential equation of the form : � Using the method of variation of parameters, solve [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 d2 y the differential equation + a2y = sec ax. (10) dx2 1 not an exact differential equation and is x2� y 2 � Find the general solution of the equation (15) an integrating factor for it. Hence solve this 2 dy 2 d y 2 2 2 2 2 x + x + y = ln x sin (ln x). differential equation for f(x +y ) = (x +y ) . (10) dx2 dx � Find the curve for which the part of the tangent � By using Laplace transform method, solve the cut-off by the axes is bisected at the point of differential equation (D2 + n2) x = a sin (nt + �), tangency. (10) d 2 � Solve by the method of variation of parameters : D2 = subject to the initial conditions x = 0 and dt 2 dy –5y = sin x (10) dx dx = 0, at t = 0, in which a, n and � are constants. dt � Solve the differential equation : (20) (15) 3 2 3 d y2 d y dy x3 �3 x2 � x �8 y � 65cos(loge x ) dx dx dx 2012 � Solve the following differential equation : (15) ()x y 2 2 dy2 xy e d y dy 2x � Solve � 2 2 (12) x– 2( x� 1) � ( x � 2) y � ( x – 2) e , dx y2(1� e (x y ) ) � 2 x 2 e (x y ) dx2 dx when ex is a solution to its corresponding � Find the orthogonal trajectories of the family of homogeneous differential equation. curves x2� y 2 � ax. (12)

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 IAS PREVIOUS YEARS QUESTIONS (2017–1983) ORDINARY DIFFERENTIAL EQUATIONS / 3 � Using Laplace transforms, solve the intial value 2010 problem y��2 y � � y � e�t ,(0) y � � 1,(0)1 y� � � Consider the differential equation (12) y� �� x, x � 0 � Show that the differential equation where � is a constant. Show that– (2xy log y ) dx�� x2 � y 2 y 2 �1� dy � 0 (i) if �(x) is any solution and �(x) = �(x) e–�x, then �(x) is a constant; is not exact. Find an integrating factor and hence, (ii) if ��< 0, then every solution tends to zero as the solution of the equation. (20) x � � . (12) � Find the general solution of the equation � Show that the diffrential equation

2 y�� y �� 12 x � 6 x . (20) (3y2 – x )� 2 y ( y2 � 3 x ) y� � 0 � Solve the ordinary differential equation admits an integrating factor which is a function of 2 2 (x+y ). Hence solve the equation. (12) x( x� 1) y�� (2 x � 1) y � � 2 y � x (2 x � 3) (20) � Verify that

1 1 2011 x �Mx� Ny � d �log ( xy ) � � ( Mx � Ny ) d (log (y )) 2 e 2 e � Obtain the soluton of the ordinary differential = M dx + N dy

dy 2 Hence show that– equation �(4x � y � 1) , if y(0) = 1. (10) dx (i) if the differential equation M dx + N dy = 0 is homogeneous, then (Mx + Ny) is an integrating � Determine the orthogonal trajectory of a family of curves represented by the polar equation factor unless Mx� Ny � 0; r = a(1 – cosθ), (r, θ) being the plane polar (ii) if the differential equation coordinates of any point. (10) Mdx� Ndy � 0 is not exact but is of the form

� Obtain Clairaut’s orm of the differential equation f1 ( xy ) y dx� f2 ( xy ) x dy � 0

–1 �dy �� dy � 2 dy then (Mx – Ny) is an integrating factor unless �x� y �� y � y � � a . Also find its �dx �� dx � dx Mx� Ny � 0 . (20)

general solution. (15) � Show that the set of solutions of the homogeneous � Obtain the general solution of the second order linear differential equation ordinary differential equation y� � p( x ) y � 0

x y��2 y � � 2 y � x � e cos x , where dashes denote on an interval I� [,] a b forms a vector subspace derivatives w.r. to x. (15) W of the real vector space of continous functions � Using the method of variation of parameters, solve on I. what is the dimension of W?. (20) the second order differedifferential equation � Use the method of undetermined coefficiens to find d2 y the particular solution of y�� y �sin x � (1 � x2 ) ex 2 �4y � tan 2 x . (15) dx and hence find its general solution. (20)

� Use Laplace transform method to solve the following initial value problem: 2009

2 � Find the Wronskian of the set of functions d x dx t dx 2 �2 �x � e , x (0) � 2 and � � 1 (15) dt dt dt 3 3 t�0 �3x ,| 3 x |�

on the interval [–1, 1] and determine whether the 3 2 � Obtain the general solution of �DDD– 6� 12 –8� set is linearly dependent on [–1, 1]. (12) � �

� Find the differential equation of the family of circles �2x9 – x � d y�12� e � e � , where D � . (15) in the xy-plane passing through (–1, 1) and (1, 1). �4 � dx (20) d2 y dy � Fidn the inverse Laplace transform of � Solve the equation 2x2 � 3 x – 3 y � x3 . (15) dx2 dx �s �1 � F( s )� ln� � . (20) � Use the method of variation of parameters to find �s � 5 � the general solution of the equation

2 2 dy y() x� y d y dy x � Solve: � ,y (0)� 1. (20) �3 � 2y � 2 e . (15) dx3 xy2 – x 2 y – 4 y 3 dx2 dx

2008 2006

� Solve the differential equation � Find the family of curves whose tangents form an � angle 4 with the hyperbolas xy=c, c > 0. (12) ydx�� x � x3 y 2 � dy � 0 . (12) � Solve the differential eqaution � Use the method of variation of parameters to find –1 xy2 � ex3 dx– x2 y dy � 0 the general solution of x2 y��– 4 xy � � 6 y � – x4 sin x . � � . (12) (12) 2 – tan –1 y dy � Using Laplace transform, solve the initial value � Solve �1�y � � x – e � 0 .(15) � � dx problem y��– 3 y � � 2 y � 4 t � e3t with y(0) = 1, � Solve the equation x2p2 + yp (2x + y) + y2 =0 using y�(0)� � 1. (15) the substituion y = u and xy=v and find its singular dy � Solve the differential equation solution, where p � . (15) dx x3 y��– 3 x2 y � � xy � sin(ln x ) � 1. (15) � Solve the differential equation

2 dy 3 2 � Solve the equation y�2 xp � yp � 0 where p � . 2 d y d y y �1 � dx x3 �2 x 2 � 2 � 10� 1 � 2 �. (15) dx dx x� x � (15) � Solve the differential equation

2007 d � D2 – 2 D� 2� y � ex tan x , where D � , � Solve the ordinary differential equation dx by the method of variation of parameters. (15) dy 1 � cos3x –3y sin3 x� sin6 x � sin2 3 x , 0 �x � . dx 2 2 2005 (12) � Find the solution of the equation � Find the orthogonal trejectory of a system of co- axial circles x2+y2+2gx+c=0, where g is the parameter. dy �xy2 dx � –4 xdx . (12) (12) y dy 2 2 2 2 � Solve xy� x � y � x y �1 . ( 12) � Determine the general and singular solutions of the dx

1 2 – 2 dy dy�� dy � � � Solve the differential equation (x+1)4 D3+2 (x+1)3 equation y� x � a �1 � � � � ‘a’ being a dx dx�� dx � � � � 1 D2–(x+1)2 D +(x+1)y = . (15) constant. (15) x �1

2 2 2 � Solve the differential equation (x +y ) (1+p) –2 2 � Solve �1�x � y�� 1 � x � y � � y �sin 2� log� 1 �x � � . dy (x+y) (1+p) (x+yp) + (x+yp)2 = 0, where p � , by (15) dx � Solve the differential equation reducing it to Clairaut’s form by using suitable substitution. (15) x2 y��4 xy � � 6 y � x4 sec 2 x � Solve the differential equation (sin x–x cos x) by variation of parameters. (15) y��– x sin xy � � y sin x � 0 given that y� sin x is a solution of this equation. (15) 2002 � Solve the differential equation dy � Solve x�3 y � x3 y 2 . (12) x2 y��– 2 xy � � 2 y � x log x , x � 0 dx

by variation of parameters. (15) � Find the values of �� for which all solutions of

d2 y dy 2004 x2 �3 x –� y � 0 tend to zero as x��. (12) dx2 dx � Find the solution of the following differential � Find the value of constant � such that the following dy 1 equation �ycos x � sin 2 x . (12) differential equation becomes exact. dx 2 y 2 dy 2 y � Solve y(xy+2x2y2) dx + x(xy –x2y2) dy = 0. (12) �2xe� 3 y � �� 3 x �� e � � 0 dx 4 2 x � Solve � D�4 D � 5� y � e ( x � cos x ) .(15) Further, for this value of �, solve the equation.(15)

� Reduce the equation (px–y) (py+x) = 2p where dy x� y � 4 � Solve � . (15) dy dx x� y � 6 p � to Clairaut’s equation and hence solve it. dx � Using the method of variation of parameters, find (15) d2 y dy d2 y dy �2 �y � xex sin x � x the solution of 2 with Solve (x+2) 2 – (2x� 5) � 2 y � ( x � 1) e .(15) dx dx dx dx � Solve the following differential equation �dy � y(0) = 0 and � � � 0 . (15) dx d2 y dy � �x�0 (1�x2 ) � 4 x � (1 � x2 ) y � x .(15) dx2 dx d � Solve (D–1) (D2–2 D+2) y = ex where D� .(15) dx 2003 � Show that the orthogonal trajectory of a system of 2001 confocal ellipses is self orthogonal. (12) � A continuous function y(t) satisfies the differential dy x � Solve x� ylog y � xye . (12) equation dx

d 1�t 5 x 3 dy �1�e ,0 � t � 1 � Solve (D –D) y = 4 (e +cos x + x ), where D � . � dx � 2 dt �2� 2t � 3 t ,1 � t � 5 (15) � Solve the differential equation (px2 + y2) (px + y) = If y(0) = –e, find y(2). (12) dy 2 p � (p + 1) where , by reducing it to Clairaut’ss 2 dx 2 d y dy 2 � Solve x �x �3 y � x log x . (12) form using suitable substitutions. (15) dx2 dx e

2 2 dy y y(log y ) d y 2 � Solve �log y � e . (15) differential equation �a y � sec( ax ) . dx x e x2 dx2

� Find the general solution of ayp2+(2x–b) p–y=0, a>o. (15) 1998

� Solve (D2+1)2 y = 24x cos x dy 3 �x2 � Solve the differential equation xy� � y e given that y=Dy=D2y=0 and D3y = 12 when x = 0. dx (15) � Show that the equation (4x+3y+1) dx + (3x+2y+1) � Using the method of variation of parameters, solve dy = 0 represents a family of hyperbolas having as

2 asymptotes the lines x+y = 0; 2x+y+1=0. (1992) d y �4y � 4 tan 2 x . (15) dx2 � Solve the differential equation y = 3px + 4p2.

2 d y dy 4x 2 � Solve – 5� 6y � e ( x � 9) . 2000 dx2 dx

d2 y dy � Solve the differential equation � Show that 3� 4x – 8 y � 0 has an integral dx2 dx d2 y dy which is a polynomial in x. Deduce the general 2 �2 �y � x sin x . dx dx solution. (12) d2 y dy 1997 � Reduce �P �Q y � R , where P, Q, R are dx2 dx dy x functions of x, to the normal form. � Solve the initial value problem � , dx x2 y� y 3 2 d y dy 2 Hence solve – 4x� (4 x2 –1) y � –3 ex sin2 x . y(0)=0. dx2 dx 2 2 2 2 (15) � Solve (x –y +3x–y) dx + (x –y +x–3y)dy = 0. � Solve the differential equation y = x–2a p+ap2. Fnd 4 3 2 d y d y d y dy �2x � Solve �6 � 11 � 6 � 20e sin x the singular solution and interpret it geometrically. dx4 dx 3 dx dx (15) � Make use of the transformation y(x) = u(x) sec x to � Show that (4x+3y+1)dx+(3x+2y+1) dy = 0 represents a family of hyperbolas with a common axis and obtain the solution of y��– 2 y � tan x� 5 y � 0 ; tangent at the vertex. (15) y(0)=0; y�(0)� 6 .

2 dy �d y � 2 � Solve x– y� ( x –1) �2 �x �1 � by the d y dy dx dx � Solve (1+2x)2 – 6 (1+2x) �16y = 8 (1+2x)2; � � dx2 dx method of variation of parameters. (15) y(0)� 0 and y� (0) � 2 . 1999 1996 � Solve the differential equation

1 �dy � 2 2 2 � Solve x2 (y–px) = yp2; �p � � . xdx� ydy�1 � x � y � �dx � � �2 2 � xdy� ydx� x� y � � Solve y sin 2x dx – (1+y2 + cos2x) dy = 0.

3 2 d y d y dy x 2 � – 3� 4 – 2y � e � cos x . d y dy Solve 3 2 � Solve + 2 +10y+37 sin 3x = 0. Find the value dx dx dx dx2 dx

� By the method of variation of parameters solve the � of y when x � 2 , if it is given that y=3 and

dy � 0 g when x=0. x� a �� a� � a �cos t . dx b d4 y d3 y d2 y 2 2x � Solve 4 � 23 – 3 2 = x + 3e + 4 sin x. d dx dx dx � Solve (D2–4D+4) y = 8x2e2x sin 2x, where, D � . dx 3 2 3 d y2 d y dy � Solve x3 �3 x2 � x � y � x � log x . dx dx dx 1993

1995 � Show that the system of confocal conics

2 2 � Solve (2x2+3y2–7)xdx– (3x2+2y2–8) y dy = 0. x y � �1 is self orthogonal. a2 �� b2 � � � Test whether the equation (x+y)2 dx – (y2–2xy–x2) dy = 0 is exact and hence solve it. � �1 � � � Solve y 1� � cos y dx + x�log x – x sin y 3 2 1 � � � � � � 3 d y 2 d y � � � �x � � � Solve x + 2x +2y =10 �x � � . (1998) dx3 dx2 �x � dy = 0. � Determine all real valued solutions of the equation 2 d y 2 � � w y dy Solve 2 0 = a coswt and discuss the nature y��– iy �� y � � iy � 0 , y� � . dx dx

of solution as w� w0. � Find the solution of the equation y�� 4 y � 8cos2 x

– x �3x � 4 2 2 given that y � 0 and y� � 2 when x = 0. � Solve (D +D +1) y = e cos� � . �2 �

1994 1992

dy � By eliminating the constants a, b obtain the � Solve + x sin 2y = x3cos2y.. dx differential equation of which xy = aex +be–x +x2 is a solution. 1��P � Q � – � Show that if � � is a function of x only � Find the orthogonal trajectories of the family of Q��y � x � semicubical parabolas ay2=x3, where a is a variable f() x dx say, f(x), then F() x� e� is an integrating factor parameter.

of Pdx + Qdy = 0. � Show that (4x+3y+1) dx + (3x+2y+1) dy = 0 � Find the family of curves whose tangent form angle represents hyperbolas having the following lines

� as asymptotes 4 with the hyperbola xy = c. � Transform the differential equation x+y = 0, 2x+y+1 = 0. (1998) � Solve the following differential equation y (1+xy) d2 y dy cosx� sin x � 2cos y3 x � 2cos5 x into one dx2 dx dx+x (1–xy) dy = 0. having z an independent variable where z = sin x � Solve the following differential equation (D2+4) y = and solve it. dy 2 sin 2x given that when x = 0 then y = 0 and � 2 . d x g dx � If + (x – a )� 0, (a, b and g being positive constants) dt2 b � Solve (D3–1)y = xex + cos2x. dx and x = a’ and � 0 when t=0, show that � Solve (x2D2+xD–4) y = x2. dt

1991 dy y (w.r.t ‘t’), then one can solve � f ()x , for any dx � If the equation Mdx + Ndy = 0 is of the form f (xy). 1 given f. Hence or otherwise. 1 ydx + f (xy). x dy = 0, then show that is dy x�3 y � 2 2 Mx– Ny � � 0 dx3 x� y � 6 an integrating factor provided Mx–Ny � 0. d2 y � Solve the differential equation. –1 2 2 � Verify that y = (sin x) is a solution of (1–x ) 2 (x2–2x+2y2) dx + 2xy dy = 0. dx � Given that the differential equation (2x2y2 +y) dx – dy �x � 2 . Find also the most general solution. (x3y–3x) dy = 0 has an integrating factor of the form dx xh yk, find its general solution.

4 1989 d y 4 � Solve –m y� sin mx . dx4 � Find the value of y which satisfies the equation � Solve the differential equation dy (xy3–y3–x2ex) + 3xy2 =0; given that y=1 when x = dx d4 y d3 y d2 y dy – 2� 5 – 8� 4y � ex . 1. dx4 dx3 dx2 dx � Prove that the differential equation of all parabolas � Solve the differential equation – 2 d� d2 y � 3 2 d y dy lying in a plane is �2 � = 0. – 4 – 5y� xe– x , given that y = 0 and dx� dx � dx2 dx � Solve the differential equation dy � 0 , when x = 0. 3 2 dx d y d y dy 2 �– 6 � 1 � x . dx3 dx 2 dx 1990 1988 � n n–1 If the equation � +a1� +...... + an = 0 (in d2 y dy unknown �) has distinct roots � ,� , ...... � . � Solve the differential equation –2 = 2ex 1 2 n dx2 dx Show that the constant coefficients of differential sin x. equation � Show that the equation (12x+7y+1) dx + (7x+4y+1) dn y dn�1 y dy dy = 0 represents a family of curves having as � a +...... + a �a � b has the dxn 1 dxn�1 n–1 dx n asymptotes the lines 3x+2y–1=0, 2x+y+1=0. most general solution of the form � Obtain the differential equation of all circles in a

� x � x � x 2 y = c (x) + c e 1 + c e 2 + ...... + c e n . 3 2 2 0 1 2 n d y ��dy � � dy� d y � plane n the form 1� –3� 0. where c , c ...... c are parameters. what is c (x)? 3 �� 2 � 1 2 n 0 dx ��dx � � dx� dx � � Analyse the situation where the ��– equation in (a) has repeated roots. 1987 � Solve the differential equation d2 y dy d2 y dy � Solve the equation x + (1–x) = y + ex x2 �2 x � y � 0 is explicit form. If your dx2 dx dx2 dx p–1 q–1 answer contains imaginary quantities, recast it in a � If f(t) = t , g(t) = t for t > 0 but f(t) = g(t) = 0 form free of those. for t � 0 , and h(t) = f * g, the convolution of f, g �()()p � q 1 show that h() t � tp� q �1; t � 0 and p, q are � Show that if the function can be integrated t� f() t �()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

t �d � equation x��2 x � � x � e ,� ' � � such that �dt � x(0)� 2, x� (0) � � 1.

� 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 show that L (f * g) = L(f). L(g). � Find the Laplace transform of the function

� 1 2n� � t � (2 n � 1)� f() t � � �–1 (2n+1) � � t � (2n+2)� n = 0, 1, 2, ......

1984

d2 y � Solve + y = sec x. dx2

u � Using the transformation y � , solve the xk equation x y�� + (1+2k) y� +xy = 0.

� Solve the equation (D2 � 1) x � t cos 2 t ,

given that x0� x 1 � 0 by the method of Laplace transform.

1983

2 d y dy 2 � Solve x�( x � 1) – y � x . dx dx

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

d2 y dy � Solve the equation �2 �y � t by the dt2 dt method of Laplace transform, given that y = –3 when t = 0, y = –1 when t = 1.

2017 2 2 2 2 dy y� 2pxy � p� x � 1 � � m , p � . (10) d dx � Solve (2D3 – 7D2 + 7D – 2) y = e–8x where D.� dx � Solve the differential equation (10) (8) dy 2 � Solve the differential equation �y � y� sin x � cos x� . dx d2 y dy x2 � 2x � 4y � x 4 . (8) dx2 dx 2015 � Solve the differential equation � Reduce the differential equation

2 �dy � dy 2 dy � � �2 � � ycot x � y . (15) x2p2+yp(2x+y)+y2=0, p � to Clairaut’s form. �dx � dx dx Hence, find the singular solution of the equation. � Solve the differential equation (8) 3 3x �dy � � dy � 2y � Solve the differential equation e�� 1 � � � � e � 0. (10) �dx � � dx � 2 2 d y dy 1 x2 �3 x � y � 2 (8) d2 y dx dx(1� x ) � Solve �4y � tan 2x by using the method of dx 2 d2 y dy � Solve x � �4x3 y � 8 x 3 sin x 2 by changing variation of parameter. (10) dx2 dx the independent variable. (10) 2016 4 2 � x/2 �x 3 � � Obtain the curve which passes through (1, 2) and � Solve (D� D � 1) y � e cos� � , �2 � �2xy has a slope � . Obtain one asymptote to 2 d x� 1 where D � . (10) the curve. (8) dx � Solve the dE to get the particular integral of 2014 4 2 d y d y 2 �2 � y � x cos x . (8) � Solve the differential equation dx4 dx 2 dy � Using the method of variation of parameters, solve y�2 px � p2 y , p � dx 2 2 d y dy 2 x and obtain the non-singular solution (8) x2 � x � y � x e . (10) dx dx � Solve � Obtain the singular solution of the differential d4 y equation �16y � x4 � sin x . (8) dx4

� Solve the following differential equation 2 dy� dy � � Solve x� y � � � (10) dy2 y x3 y dx� dx � � � � x tan . (10) dx x y x2 d2 y dy � Solve x2 �3 x � y � (1 � x )�2 (10) � Solve by the method of variation of parameters dx2 dx y��3 y � � 2 y � x � cos x . (10) � Solve the D.E. 2011 � Find the family of curves whose tangents form an d3 y d 2 y dy �3 � 4 � 2y � ex � cos x . (10) angle � / 4 With hyperbolas xy� c. (10) dx3 dx 2 dx 2 d y dy x � Solve 2 �2 tanx � 5 y � sec x . e . (10) 2013 dx dx dy � Solve � Solve p2 �2 py cot x � y2 Where p � . (10) dx dy � � Solve �xsin 2 y � x3 cos 2 y (8) dx 2 �x4 D 4�6 x 3 D 3 � 9 x 2 D 2 � 3 xD � 1� y �� 1 � log x � , � Solve the differential equation d 2 Where D � . (15) d y dy 2 �4x � (4 x2 � 1) y � � 3 ex sin 2 x dx dx2 dx �� Solve (D4� D 2 � 1) y � ax2 � be� x sin 2 x , where by changing the dependent variable. (13) d � Solve D � (15) dx x �3 � (D3 � 1) y � e2 sin x � � 2010 �2 � � Show that cos� x� y� is an integrating factor of d where D � . (13) dx y dx�� y �tan � x � y�� dy � 0. � Apply the method of variation of parameters to Hence solve it (8) solve d2 y dy 2 x d y � Solve 2 �2 �y � xesin x (8) �y �2(1 � ex )�1 (13) dx dx dx2 � Solve the following differential equation

dy 2012 �sin2 �x � y � 6 � (8) dx

dytan y x � Find the general solution of � Solve � �(1 � x ) e sec y . (8) dx1� x d2 y dy � Solve and find the singular solution of �2x � x2 � 1 y � 0 (12) dx2 dx � � 3 2 2 3 x p� x py � a � 0 (8) � Solve

2 2 d2 y dy 2 2 2 � � �d � � d � x � Solve: x y2 �� x � y � � 0 (10) �1 � 1 y � x � e dx dx � � �2 � (10) � � �dx � � dx �

d4 y d2 y � Solve by the method of variation of parameters the � Solve �2 �y � x2 cos x . (10) dx4 dx2 following eqaution

2 2 d y dy 2 d y dy x2 �1 � 2x � 2 y � x2 � 1 (10) �cotx �� 1 � cotx � y � ex sin x . (10) � � dx2 dx � � dx2 dx

2009 2007 � Find the orthogonal trajectories of the family of the dy � Solve sec2 y� 2 x tan y � x3 (10) dx x2 y 2 curves � �1, � being a parameter.. (10) a2 b 2 �� Find the 2nd order ODE for which ex and x2 ex 2x 3x are solutions. (10) � Show that e and e are linearly independent

2 3 2 2 3 d y dy � Solve � y�2 yx� dx �� 2 xy � x� dy � 0 . (10) solutions of �5 � 6y � 0. Find the general dx2 dx

2 dy � �dy � dy solution when y�0 � � 0 and �1 (10) � Solve � � �2 coshx � 1 � 0. (8) dx � �dx � dx �0 � Find the family of curves whose tangents form an d3 y d 2 y dy � Solve �3 � 3 �y � x2 e� x (10) angle � � 4 with the hyperbola xy� c. (10) dx3 dx 2 dx � Apply the method of variation of parametes to

2 � Show that ex is a solution of solve � D2� a 2 � y � cos ec ax (10)

2 d y dy 2 2 2 d y 2 �4x �� 4 x � 2 � y � 0. (12) � Find the general solution of 1�x � 2 x dx dx � � dx2 Find a second independent solution. dy �3y � 0 solution of it. (10) dx 2008

2 2006 � Show that the functions y1 � x� � x and

2 2 2 � From x� y �2 ax � 2 by � c � 0, derive differential y2 � x� � xloge x are linearly independent obtain equation not containing, a,. b or c (10) the differential equation that has y1 � x� and y2 � x� � Discuss the solution of the differential equation as the independent solutions. (10) � Solve the following ordinary differential equation �2 � 2 �dy � 2 of the second degree : y ��1 �� a (10) ��dx � � 2 �dy � dy y� � ��2 x � 3 � � y � 0 (10) d2 y dy �dx � dx � Solve x��1 � x � � y � ex (10) dx2 dx �dy �� dy � dy � Reduce the equation x� y x � y � 2 d4 y �dx �� dx � dx � Solve �y � xsin x (10) � �� dx4 to clairaut’s form and obtain there by the singular integral of the above equation. (10) d2 y dy � Solve x2 � x � y � x2 ex (10/2008) � Solve dx2 dx

2 � Reduce 2 d y dy �1�x � 2 �� 1 � x � � y � 4cosloge � 1 � x � (10) 2 dx dx �dy � 2 2 dy xy� � �� x � y �1 � � xy � 0 � Find the general solution of the equation �dx � dx

to clairaut’s form and find its singular solution. 2 � Solve 1�x y " � 1 � x y ' � y � 4cos ln 1 � x (10) � � � � � � �� ,y� 0� � 1, y� e � 1� � cos1. (10) 2005 � Obtain the general solution of � Form the differential equation that represents all �x 2 (10) parabolas each of which has latus rectum 4a and y"� 2 y ' � 2 y � 4 e x sinx . Whose are parallel to the x-axis. (10) � Find the general solution of � xy3 � y� dx � 2 � (i) The auxiliary polynomial of a certain homogenous linear differential equation with � x2 y 2 � x � y4 � dy � 0 (10) constant coefficients in factored form is 4 3 2 3 � Obtain the general solution of �DDDD�2 � � 2 � P� m �� m4 � m � 2 �6 �m2 �6 m � 25� . 2x dy What is the order of the differential equation y� x � e , Where Dy � . (10) dx and write a general solution ? (ii) Find the equation of the one-parameter family 2003 of parabolas given by y2 �2 cx � c2 , C real � Find the orthogonal trajectories of the family of co- and show that this family is self-orthogonal. 2 2 g (10) axial circles x� y �2 gx � c � 0 Where is a � Solve and examine for singular solution the parameter. (10)

2 2 2 2 2 d3 y d 2 y dy following equation P� x� a� �2 pxy � y � b � 0 � Find the three solutions of �2 � � 2y � 0 dx3 dx 2 dx (10) Which are linealy independent on every real d2 y interval. (10) � Solve the differential equation �9y � sec3 x dx2 � Solve and examine for singular solution: (10) y2 �2 pxy � p2� x 2 �1� � m2 . (10)

1 3 2 � Given y� x � is one solution solve the 3 d y2 d y �1 � x � Solve x3 �2 x2 � 2 y � 10� x � � (10) dx dx �x � d2 y dy differential equation x2 � x � y � 0 � Given y� x is one solutions of dx2 dx 2 reduction of order method. (10) 3 d y dy �x �1 � 2 � 2x � 2 y � 0 find another linearly � Find the general solution of the defferential dx dx independent solution by reducing order and write d2 y dy equation �2y � 3 y � 2 ex � 10sin x by the the general solution. (10) dx2 dx method of undertermined coefficients. (10) � Solve by the method of variation of parameters

d2 y 2004 �a2 y � sec ax , a real. (10) dx2 � Dertermine the family of orthgonal trajectories of the family y� x � ce�x (10) 2002

� Show that the solution curve satisfying 4 � If �D� a � enx is denoted by z , prove that 2 � x� xy� y' � y3 Where y�1 as x � � , is a �z �2 z � 3 z z ,, all vanish when n� a . Hence conic section. Indentify the curve. (10) �n �n2 � n 3 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 14 / ORDINARY DIFFERENTIAL EQUATIONS IFoS PREVIOUS YEARS QUESTIONS (2017–2000) � Using differential equations show that the system show that enx,,, xe nx x2 e nx x� nx are all solutions of x2 y2 4 d of confocal conics given by 2 �2 � 1, �D� a � y � 0 . Here D Stands for . (10) a�� b � � dx is self othogonal. (10)

2 2 2 � Solve 4xp�� 3 x � 1 � � 0 and examine for singular 2 d y dy 2 � Solve �1�x � 2 � x � a y � 0 given that solutions and extraneous loci. Interpret the results dx dx

�1 geometrically. (10) y� easin x is one solution of this equation. (10) � (i) Form the differential equation whose primitive is n � Find a general solution y� y � tan x , ��x � � � cos x � � sin x � 2 2 y� A�sin x �� B �cos x � � by variation of parameters. (10) � x � � x � (ii) Prove that the orthogonal trajectory of system 2000 of parabolas belongs to the system itself. (10) � Using variation of parameters solve the differential � Solve � xy2� 2 ��1 � P �2 � 2 � xy � �� 1 � pxyp �� xyp � �2 �0 equation dy P � . Interpret geometrically the factors in the P- 2 dx d y dy 2 x2 2 �4x �� 4 x � 1 � y � � 3 e sin 2 x . (10) dx dx and C-discriminants of the equation 8p3 x� y� 12 p 2 � 9� � (i) Solve the equation by finding an integrating (20) factor � Solve

of � x�2� sin ydx � x cos ydy � 0. 2 2 d y2 dy a �i � 2 � �4 y � 0 (ii) Verify that � � x� � x2 is a solution of dxx dx x d2 y dy 2 ii�tan x � 3cos x � 2 y cos2 x � cos 4 x . y"� y � 0 and find a second independent � � 2 � � x2 dx dx solution. (10) by varying parameters. (20/2007) � Show that the solution of � D2n� 1 �1� y � 0,

consists of Aex and n paris of terms of the form 2� r eax � bcos� x� c sin � x� , Where a � cos r r 2n � 1 2� r and � � sin ,r� 1,2....., n and b, c are 2n � 1 r r arbitrary constants. (10)

2001 � A constant coefficient differential equation has auxiliary eqution expressible in factored form as

2 2 P� m �� m3 � m �1 � � m2 � 2 m � 5� . What is the order of the differential equation and find its general solution. (10)

2 2 �dy � dy 2 � Solve x� � � y�2 x � y � � y � 0 (10) �dx � dx