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 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 / ORDINARY DIFFERENTIAL EQUATIONS IAS ­ PREVIOUS YEARS QUESTIONS (2017–1983) � 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 |� HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi­60.