Partial Differential Equations

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1 Partial Differential Equations Previous ear Questions from 016 to 199 Ramanasri 017 S H O P NO- 4, 1 S T F L O O R, N E A R R A P I D F L O U R M I L L S, O L D R A J E N D E R N A G A R, N E W D E L H I. W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : /6363/6464 1

2 Find the general equation of surfaces orthogonal to the famil of spheres given b z cz. (10 marks). Final he general integral of the particle differential equation ( z) p ( z) q (10 marks) 3. Determine the characteristics of the equation z p q find the integral surface which passes though the parabola 4z 0 (15 marks) 4. Solve the particle differential equation 3 z 3 z 3 z 3 z e (15 marks) Find the temperature u(, t) 3 in a bar of silver of length constant cross section of area densit p 10.6 g / cm, thermal conductivit K 1.04 / ( cmsec C) specific heat / gcthe bar is perfectl isolated laterall with ends kept at 0 C f( ) sin(0.1 ) C note that u(, t) follows the head equation 015 u t c u 1cm. Let initial temperature where c k/ ( ) (0 marks) 6. z z Solve the partial differential equation: ( z ) p q z 0 where p q 7. Solve : ( ' ' ) D DD D u e, where D D' 8. Solve for the general solution pcos( ) qsin( ) z, where 9. Find the solution of the initial-boundar value problem u u u 0, 0 l, t 0 t u(0, t) u( l, t) 0, t 0 z z p q u(, 0) ( l ), 0 l 10. Reduce the second-order partial differential equation u u u u u 0into canonical form. Hence, find its general solution Solve the partial differential equation (D 5 DD' D' ) z 4( ) 1. Reduce the equation z z to canonical form. 13. Find the deflection of a vibrating string (length=, ends fied, initial velocit initial deflection. f( ) k(sin sin ) u u ) corresponding to zero t Reputed Institute for IAS, IFoS Eams Page

3 u u 14. Solve, 0 1, t 0, given that t (i) u(, 0) 0, 0 1; (ii) u (, 0), 0 1 t (iii) u(0, t) u(1, t) 0, for all t From a partial differential equation b eliminating the arbitrar functions z f( ) g( ) f g from z z z 16. Reduce the equation ( ) 0 to its canonical from when 17. Solve ( D DD' 6 D' ) z sin( ) where D D ' denote 18. Find the surface which intersects the surfaces of the sstem z( ) C(3z 1), ( C being a constant ) orthogonall which passes through the circle 1, z A tightl stretched string with fied end points l is initiall at rest in equilibrium position. If it is set vibrating b giving each point a velocit. ( l ), find the displacement of the string at an distance from one end at an time t Solve partial differential equation ( D D')( D D') z e (1 Marks) 1. Solve partial differential equation p q 3z. A string of length l is fied at its ends. The string from the mid-point is pulled up to a height k then released from rest. Find the deflection (, t ) of the vibrating string. 3. The edge r a of a circular plate is kept at temperature f( ). The plate is insulated so that there is no loss of heat from either surface. Find the temperature distribution in stead state. 011 ( ) 4. Solve the PDE ( D D' D 3 D' ) z e (1 Marks) z z 5. Solve the PDE ( z) (4 z ) 6. Find the surface satisfing 1 0. z 6 touching (1 Marks) 3 3 z along its section b the plane u u 7. Solve 0, 0 a, 0 b satisfing the boundar conditions u 3 u(0, ) 0, u(, 0) 0, u(, b) 0 ( a, ) T sin a Reputed Institute for IAS, IFoS Eams Page 3

4 (, t) 8. Obtain temperature distribution in a uniform bar of unit length whose one end is kept at the other end is insulated. Also it is given that (, 0) 1, Solve the PDE ( D D')( D D') Z e (1 Marks) 30. Find the surface satisfing the PDE ( D DD' D' ) Z 0 the conditions that 0 bz when 0 (1 Marks) 31. Solve the following partial differential equation az zp q 0( s) s, 0( s) 1, z0( s) s b the method of characteristics. 3. Reduce the following nd order partial differential equation into canonical form find its general solution. u u u Solve the following heat equation u u 0, 0, t 0 t u(0, t) u(, t) 0 t 0 u(, 0) ( ), when 34. Show that the differential equation of all cones which have their verte at the origin is p q z. Verif that this equation is satisfied b the surface z z 0. (1 Marks) 35. (i) Form the partial differential equation b elimination the arbitrar function given b: (ii) f(, z ) 0 Find the integral surface of:, z 1 p p z 0 which passes through the curve: 36. Find the characteristics of: r t 0 where r t have their usual meanings. 37. Solve : ( ' ' ) ( )sin cos D DD D z where D f D ' represent 38. A tightl stretched string has its ends fied at 0 1. At time t 0, the string is given a shape defined b f( ) ( l ), where is a constant, then released. Find the displacement of an point t. of the string at time Find the general solution of the partial differential equation( 1) p ( z ) q ( z) also find the particular solution which passes through the lines 1, 0 (1 Marks) Reputed Institute for IAS, IFoS Eams Page 4

5 40. Find the general solution of the partial differential equation: ( D DD' 6 D' ) z cos, where D, D' (1 Marks) 41. Find the stead state temperature distribution in a thin rectangular plate bounded b the lines 0, a, 0 b. The edges 0, a the edge b is kept at C. 0 are kept at temperature zero while 4. Find complete singular integrals of z p q pq 0 using Charpit s method. 43. z z Reduce canonical form (i) Form a partial differential equation b eliminating the function 1 z f log (ii) Solve z p q pq 0 (6+6=1 Marks) 45. Transform the equation z z 0 into one in polar coordinates thereb show that the solution of the given equation represents surfaces of revolution. (1 Marks) 46. Solve u u 0 in D where D (, ) : 0 a,0 b is a rectangle in a plane with the boundar conditions: u(,0) 0, u(, b) 0, 0 a u(0, ) g( ), u ( a, ) h( ), 0 b. Reputed Institute for IAS, IFoS Eams Page 5 f from: u u 47. Solve the equation c b separation of variables method subject to the conditions: t u(0, t) 0 u( l, t), for all t u(,0) f( ) for all in [0, l ] 48. Solve : p( z ) ( z q)( z ) (1 Marks) z z z 49. Solve : 4 4 sin(3 ) (1 Marks) The deflection of vibrating string of length l, is governed b the partial differential equation. The ends of the string are fied at 0 l. The initial velocit is zero. The initial utt C u l, 0 l displacement is given b u (,0) 1 l l, l. l Find the deflection of the string at an instant of time. 51. Find the surface passing through the parabolas z 0, 4a satisfing the equation z z 0 z 1, 4a

6 5. Solve the equation z z p q z, p, q Formulate partial differential equation for surfaces whose tangent planes form a tetrahedron of constant volume with the coordinate planes. (1 Marks) 54. Find the particular integral of ( z) p ( z ) q z( ) which represents a surface passing through z (1 Marks) 55. The ends A B of a rod 0cm long have the temperature at 30 0 C 80 0 C until stead state prevails. The temperatures of ends are changed to 40 0 C 60 0 C respectivel. Find the temperature distribution in the rod at timet. 56. Obtain the general solution of ( D 3 D' ) z e sin( 3 ) where D D' Find the integral surface of the following partial differential equation : ( z) p ( z) q ( ) z (1 Marks) 58. Find the complete integral of the partial differential equation ( p q ) pz deduce the solution which passes through the curve 0, z 4. (1 Marks) z z z 59. Solve the partial differential equation : ( 1) e 60. A uniform string of length lwith no initial displacement, is struck at a, 0 a l, with velocit. Find the displacement of the string at an time t Using Charpit s method, find the complete solution of the partial differential equation p q z l, held tightl between 0 v z z z 6. Find the general solution of 3 cos( 3 ) (1 Marks) 63. Show that the differential equations of all cones which have their verte at the origin are p q z. Verif that z z 0 is a surface satisfing the above equation. (1 Marks) 64. Solve z z z z 3 3 e 65. Solve the equation p q p q 0 using Charpit s method. Also find the singular solution of the equation, if it eists. Reputed Institute for IAS, IFoS Eams Page 6

7 66. Find the deflection of a vibrating string, stretched between fied points (0, 0) (3, 0), corresponding to zero initial velocit following initial deflection: h when 0 1 l h(3l ) f( ) when l l l h( 3 l) when l 3l l is a constant. Whereh u(, t) 00 l 67. Find two complete integrals of the partial differential equation p q 4 0 (1 Marks) 1 ( ) ( )( ) 68. Find the solution of the equation z p q p q (1 Marks) 69. Frame the partial differential equation b eliminating the arbitrar constants a b from log( az 1) a b 70. Find the characteristic strips of the equation p q pq 0 then find the equation of the integral surface through the curve z, 0 u u 71. Solve:, 0 l, t 0 t u(0, t) u( l, t) 0 u(, 0) ( l ), 0 t Find the complete integral partial differential equation p q 3 8 q ( ) (1 Marks) z 73. Find the general integral of the equation m( ) nz z l( ) nz ( l m) z (1 Marks) 74. Prove that for the equation z p q 1 pq 0 the characteristic strips are given b 1 1 ( t ), ( t t ), z ( t ) E ( AC BD ) e t t B Ce A De t t p( t) A( B Ce ), q( t) B( A De ) where A, B, C, D E are arbitrar constants. Hence find the values of these arbitrar constants if the integral surface passes through the line z 0, 75. Write down the sstem of equations for obtaining the general equation of surfaces orthogonal to the famil given b ( z ) C 1 z z z z Solve the equation constant coefficients. b reducing it to the equation with Reputed Institute for IAS, IFoS Eams Page 7

8 77. Solve : pq z 78. Prove that if m n l when z 0, ( S ) p ( S ) p ( S ) p S z can be given in the form ( 1 ) ( ) ( 3 ) (1 Marks) the solution of the equation S z z z z where S 1 3 z z pi, i 1,,3. i (1 Marks) 79. Solve b Charpit s method the equation p ( 1) pq q ( 1) pz qz z Solve : ( ' ' ) 3 D DD D z e. 81. A tightl stretched string with fied end points 0, l is initiall at rest in equilibrium position. If it is set vibrating b giving each point of it a velocit obtain at time t the displacement at a distance from the end 0 k( l ), Verif that the differential equation ( z) d ( z z ) d ( ) dz 0 is integrable find its primitive. 83. Find the surface which intersects the surfaces of the sstem z( ) c(3z 1), is constant, orthogonall which passes through the circle 1, z Find the characteristics of the equation pq z, determine the integral surface which passes through the passes through the parabola z 85. Use Charpit s method to find a complete integral to 0,. p q p q 1 0 z z 86. Find the solution of the equation e cos which when One end of a string ( 0) is fied, the point a is made to oscillate, so that at time t the displacement is. Show that the displacement u(, t) of the point at time t is given b gt () 0 as has the value cos t a u(, t) f( ct ) f( ct ) where f is a function satisfing the relation f( t a) f( t) g c Find the differential equation of the set of all right circular cones whose aes coincide with the ais 89. Form the differential equation b eliminating ab, c from z a( ) b( ) abt c 90. u u u Solve w z z Reputed Institute for IAS, IFoS Eams Page 8 c z-

9 91. Find the integral surface of the linear partial the differential equation z z ( z) ( z) ( ) z through the straight line 0, z 1 z z 9. Use Charpit s method to find a complete integral of z 1 z 93. Find a real function which reduces to zero when satisfies the equation V (, ), 0 V V 4 ( ) 94. Appl Jacobi s method to find a complete integral of the equation z z z z (i) Find the differential equation of all surfaces of revolution having z- ais as the ais of rotation. (ii) Form the differential equation b eliminating a b from z ( a)( b) 96. Find the equation of surfaces satisfing 4zp q 0 passing through z z (10+10=0 Marks) 1, 97. Solve : ( z) p ( z ) q 98. Use Charpit s method to find complete integral of z ( p z q ) Solve : ( D D ) z 100. Appl Jacobi s method to find complete integral of z z z p, p, p z p p p Here is a function of 1,, (i) differential equation of all spheres of radius having their center in f - plane (ii) Form differential equation b eliminating g from z f( ) g( ) (10+10=0 Marks) 10. Solve : z ( p q 1) C 103. Find the integral surface of the equation ( ) p ( ) q ( ) z passing through the curve z a 3, Appl Charpit s method to find the complete integral of z p a p q 105. z z Solve : cosm cosn 106. Find a surface passing through the lines z 0 z 1 0satisfing z z z Reputed Institute for IAS, IFoS Eams Page 9

10 In the contet of a partial differential equation of the first order in there independent variables, define illustrate the terms: (i) The complete integral (ii) The singular integral 108. w w w Find the general integral of ( z w) ( z w) ( w) z z 109. Obtain the differential equation of the surfaces which are the envelopes of a one-parameter famil of planes Eplain in detail the Charpit s method of solving the nonlinear partial differential equation z z f,, z,, z z z 3 Solve z Solve ( D 7D D 6 D ) z sin( ) e Find the differential equation of the famil of all cones with verte at (, 3,1) 114. Find the integral surface of, z Obtain a Complete Solution of z z which passes through the hperbola p q z 0, p, q pq m n l z 116. Use the Charpit s method to solve16p z 9q z 4z 4 0. Interpret geometricall the complete solution mention the singular solution Solve ( D 3 DD' D' ) z, b eping the particular integral in ascending powers of D,as well as in ascending powers of D ' Find a surface satisfing ( D DD') z 0 touching the elliptic paraboloid z 4 along its section b the plane Find the surface whose tangent planes cut off an intercept of constant length R from the ais ofz Solve ( 3 ) p ( 3 ) q ( ) z 11. Find the integral surface of the partial differential equation ( ) p ( z) q z through the circle z 1, 1 1. Using Charpit s method find the complete integral of z p q pq Solve r s q z 14. Find the general solution of r t p q log Reputed Institute for IAS, IFoS Eams Page 10

11 Solve: ( z z z ) p ( z z z ) q ( z z z ) 16. Find the complete integral of ( )( q p) ( p q) 17. Use Charpit s method to solve p q z 1 pq 18. Find the surface passing through the parabolas z 0, 4 a; z 1, 4a satisfing the differential equation r p Solve : r s 6t cos 130. Solve: u z z z cos( ) e Reputed Institute for IAS, IFoS Eams Page 11

Partial Differential Equations

++++++++++ Partial Differential Equations Previous year Questions from 017 to 199 Ramanasri Institute 017 W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : 8 7 5 0 7 0 6 6 /