Partial Differential Equations

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1 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 : / 6 6 1

2 Solve ( D DD' D' ) z e sin where D, D', D, D'.. Let be a closed curve in y- plane and let S denote the region bounded by the curve. Let [10 Marks] w w (, ) (, ). f y y S If f is prescribed at each point ( y, ) of S and w is prescribed on the boundary of S then prove that any solution w w(, y), satisfying these conditions, is unique. [10 Marks]. Find a complete integral of the partial differential equation ( pq yp q) y 0. [15 Marks] 4. z z z y z z Reduce the equation y y y y to canonical form and hence solve it. [15 Marks] 5. Given the one-dimensional wave equation y y c ; t t 0, where T c, T the constant tension in the m string and m is the mass per unit length of the string. (i) Find the appropriate solution of the wave equation (ii) Find also the solution under the conditions y(0, t) 0, y( l, t) 0 for all t and t t0 0, y(,0) asin,0 l, a 0. [0 Marks] t Find the general equation of surfaces orthogonal to the family of spheres given by y z cz. (10 marks) 7. Final he general integral of the particle differential equation( y z) p ( yz) q y (10 marks) 8. Determine the characteristics of the equation z p q and find the integral surface which passes though the parabola 4z 0 (15 marks) 9. Solve the particle differential equation z z z z e (15 marks) 10. Find the temperature u(, t) in a bar of silver of length and constant cross section of area1cm. Let density p 10.6 g / cm, thermal conductivity K 1.04 / ( cmsec C) and specific heat / gc the bar is perfectly isolated laterally with ends kept at 0 C and initial temperature f( ) sin(0.1 ) C note that u(, t) follows the head equationu t c u wherec k/ ( ) (0 marks) Solve the partial differential equation: z z where p andq y ( y z ) p yq z 0 Reputed Institute for IAS, IFoS Eams Page

3 y 1. Solve : ( D DD' D' ) u e, where D and D' y 1. Solve for the general solution pcos( y) qsin( y) z, where 14. Find the solution of the initial-boundary value problem u u u 0, 0 l, t 0 t u(0, t) u( l, t) 0, t 0 z z p and q y u(, 0) ( l ), 0 l 15. Reduce the second-order partial differential equation u u u u u y y y 0into canonical form. Hence, find its general solution Solve the partial differential equation (D 5 DD' D' ) z 4( y ) 17. Reduce the equation z z to canonical form. 18. Find the deflection of a vibrating string (length=, ends fied, initial velocity and initial deflection. f( ) k(sin sin ) u u 19. Solve, 0 1, t 0, given that t (i) u(, 0) 0, 0 1; u u ) corresponding to zero t (ii) u (, 0), 0 1 t (iii) u(0, t) u(1, t) 0, for all t From a partial differential equation by eliminating the arbitrary functions f and g from z yf( ) g( y) z z z 1. Reduce the equation y ( y) 0 to its canonical from when y. Solve ( D DD' 6 D' ) z sin( y) where D and D ' denote and. Find the surface which intersects the surfaces of the system z( y) C(z 1), ( C being a constant ) orthogonally and which passes through the circle y 1, z 1 4. A tightly stretched string with fied end points 0 and l is initially at rest in equilibrium position. If it is set vibrating by giving each point a velocity. ( l ), find the displacement of the string at any distance from one end at any time t Reputed Institute for IAS, IFoS Eams Page

4 01 5. y Solve partial differential equation ( D D')( D D') z e (1 Marks) 6. Solve partial differential equation p qy z 7. A string of length l is fied at its ends. The string from the mid-point is pulled up to a height k and then released from rest. Find the deflection y(, t ) of the vibrating string. 8. 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 steady state. 011 ( ) 9. Solve the PDE ( D D' D D' ) z e y (1 Marks) z z 0. Solve the PDE ( z) (4 z y) y 1. Find the surface satisfying y1 0. z 6 and touching (1 Marks) z y along its section by the plane u u. Solve 0, 0 a, 0 y b satisfying the boundary conditions u y u(0, y) 0, u(, 0) 0, u(, b) 0 ( a, y) T sin a 0. Obtain temperature distribution y(, t ) in a uniform bar of unit length whose one end is kept at 10 and the other end is insulated. Also it is given that y(, 0) 1, Solve the PDE ( D D')( D D') Z e y (1 Marks) 5. Find the surface satisfying the PDE ( D DD' D' ) Z 0 and the conditions that bz y when 0 and az when y 0 (1 Marks) 6. Solve the following partial differential equation zp yq 0( s) s, y0( s) 1, z0( s) s by the method of characteristics. 7. Reduce the following nd order partial differential equation into canonical form and find its general solution. u u u 0 y 8. Solve the following heat equation u u 0, 0, t 0 t u(0, t) u(, t) 0 t 0 u(, 0) ( ), 0 Reputed Institute for IAS, IFoS Eams Page 4

5 Show that the differential equation of all cones which have their verte at the origin is p qy z. Verify that this equation is satisfied by the surface yz z y 0. (1 Marks) 40. (i) Form the partial differential equation by elimination the arbitrary function f given by: (ii) f( y, z y) 0 Find the integral surface of: y y, z 1 p y p z 0 which passes through the curve: 41. Find the characteristics of: y r t 0 where r and t have their usual meanings. 4. Solve : ( D DD' D' ) z ( y y )sin y cos y where D and D' represent and 4. A tightly stretched string has its ends fied at 0 and 1. At time t 0, the string is given a shape defined by f( ) ( l ), where is a constant, and then released. Find the displacement of any point of the string at timet Find the general solution of the partial differential equation(y 1) p ( z ) q ( yz) and also find the particular solution which passes through the lines 1, y 0 (1 Marks) 45. Find the general solution of the partial differential equation: ( D DD' 6 D' ) z ycos, where D, D' (1 Marks) 46. Find the steady state temperature distribution in a thin rectangular plate bounded by the lines 0, a, y 0 and y b. The edges and 0, a and y 0 are kept at temperature zero while the edge y b is kept at C. 47. Find complete and singular integrals of z p qy pq 0using Charpit s method. 48. z z Reduce canonical form (i) Form a partial differential equation by eliminating the function f from: z y f log y 1 (ii) Solve z p qy pq 0 (6+6=1 Marks) 50. Transform the equation yz z 0 into one in polar coordinates and thereby show that the y solution of the given equation represents surfaces of revolution. (1 Marks) 51. Solve u u 0 D (, y) : 0 a,0 y b is a rectangle in a plane with the in D where yy boundary conditions: Reputed Institute for IAS, IFoS Eams Page 5

6 u(,0) 0, u(, b) 0, 0 a u(0, y) g( y), u ( a, y) h( y), 0 y b. u u 5. Solve the equation c by separation of variables method subject to the conditions: t u(0, t) 0 u( l, t), for all t and u(,0) f( ) for all in [0, l ] 5. Solve : 006 p( z y ) ( z qy)( z y ) (1 Marks) z z z 54. Solve : 4 4 sin( y) (1 Marks) 55. The deflection of vibrating string of length l, is governed by the partial differential equation. The ends of the string are fied at and 0 and l. The initial velocity is zero. The initial utt C u l, 0 l displacement is given by u (,0) 1 l l, l. l Find the deflection of the string at any instant of time. 56. Find the surface passing through the parabolas z 0, y 4a and z z satisfying the equation 0 z z 57. Solve the equation p q y z, p, q 005 z 1, y 4a and 58. Formulate partial differential equation for surfaces whose tangent planes form a tetrahedron of constant volume with the coordinate planes. (1 Marks) 59. Find the particular integral of ( y z) p y( z ) q z( y) which represents a surface passing through y z (1 Marks) 60. The ends A and B of a rod 0cm long have the temperature at 0 0 C and 80 0 C until steady state prevails. The temperatures of ends are changed to 40 0 C and 60 0 C respectively. Find the temperature distribution in the rod at timet. 61. Obtain the general solution of ( D D' ) z e sin( y ) where D and D' y Find the integral surface of the following partial differential equation : ( y z) p y( z) q ( y ) z (1 Marks) Reputed Institute for IAS, IFoS Eams Page 6

7 6. Find the complete integral of the partial differential equation ( p q ) pz and deduce the solution which passes through the curve 0, z 4y. (1 Marks) z z z 64. Solve the partial differential equation : ( y1) e 65. A uniform string of length l, held tightly between 0 and lwith no initial displacement, is struck at a, 0 a l, with velocityv 0. Find the displacement of the string at any time t Using Charpit s method, find the complete solution of the partial differential equation p q y z 00 z z z 67. Find the general solution of y cos( y) (1 Marks) 68. Show that the differential equations of all cones which have their verte at the origin are p qy z. Verify that yz z y 0 is a surface satisfying the above equation. (1 Marks) z z z z 69. Solve y e 70. Solve the equation p q p qy y 0 using Charpit s method. Also find the singular solution of the equation, if it eists. 71. Find the deflection u(, t) of a vibrating string, stretched between fied points (0, 0) and ( l, 0), corresponding to zero initial velocity and following initial deflection: h when 0 1 l h(l ) f( ) when l l l h( l) when l l l Whereh is a constant Find two complete integrals of the partial differential equation p y q 4 0 (1 Marks) 1 ( ) ( )( ) 7. Find the solution of the equation z p q p q y (1 Marks) 74. Frame the partial differential equation by eliminating the arbitrary constants a and b from log( az 1) ay b 75. Find the characteristic strips of the equation p yq pq 0 and then find the equation of the integral surface through the curve z, y 0 u u 76. Solve:, 0 l, t 0 t u(0, t) u( l, t) 0 u(, 0) ( l ), 0 t. Reputed Institute for IAS, IFoS Eams Page 7

8 Find the complete integral partial differential equation p q y 8 q ( y ) (1 Marks) z 78. Find the general integral of the equation my( y) nz z l( y) nz ( l my) z (1 Marks) 79. Prove that for the equation z p qy 1 pq y 0 the characteristic strips are given by 1 1 ( t ), y ( 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 and E are arbitrary constants. Hence find the values of these arbitrary constants if the integral surface passes through the line z 0, y 80. Write down the system of equations for obtaining the general equation of surfaces orthogonal to the family given by ( y z ) C y 81. Solve the equation z z z z y y y constant coefficients. 8. Solve : 8. Prove that if pq y z m n l when z 0, the solution of the equation ( S ) p ( S ) p ( S ) p S z can be given in the form ( 1 ) ( ) ( ) 1 by reducing it to the equation with S z z z z where S 1 z and z pi, i 1,,. i (1 Marks) (1 Marks) 84. Solve by Charpit s method the equation p ( 1) pqy q y( y 1) pz qyz z Solve : ( ' ' ) y D DD D z y e. 86. A tightly stretched string with fied end points 0, l is initially at rest in equilibrium position. If it is set vibrating by giving each point of it a velocity k( l ), obtain at time t the displacement y at a distance from the end Verify that the differential equation ( y yz) d ( z z ) dy ( y y) dz 0 is integrable and find its primitive. 88. Find the surface which intersects the surfaces of the system z( y) c(z 1), c is constant, orthogonally and which passes through the circle y z 1, 1 Reputed Institute for IAS, IFoS Eams Page 8

9 89. Find the characteristics of the equation pq z, and determine the integral surface which passes through the passes through the parabola y z 90. Use Charpit s method to find a complete integral to 0,. p q p qy 1 0 z z 91. Find the solution of the equation e cos ywhich 0 as and has the value cos y when 0 9. One end of a string ( 0) is fied, and the point a is made to oscillate, so that at time t the displacement is gt (). Show that the displacement u(, t) of the point at time t is given by t a u(, t) f( ct ) f( ct ) where f is a function satisfying 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 z- ais 94. Form the differential equation by eliminating aband, c from z a( y) b( y) abt c 95. u u u Solve y w yz z 96. Find the integral surface of the linear partial the differential equation z z ( y z) y( z) ( y ) z through the straight line y 0, z 1 z z 97. Use Charpit s method to find a complete integral of z 1 z y 98. Find a real function V(, y ), which reduces to zero when y 0 and satisfies the equation V V 4 ( y ) 99. Apply 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 by eliminating a and b from z ( a)( y b) 101. Find the equation of surfaces satisfying 4yzp q y 0 and passing through (10+10=0 Marks) y z 1, z 10. Solve : ( y z) p ( z ) q y 10. Use Charpit s method to find complete integral of z ( p z q ) Solve : ( D D ) z y y Reputed Institute for IAS, IFoS Eams Page 9

10 105. Apply Jacobi s method to find complete integral of z z z p, p, p 1 1 p p p Here 1 1. and z is a function of 1,, (i) differential equation of all spheres of radius having their center in y- plane (ii) Form differential equation by eliminating f and g from z f( y) g( y) (10+10=0 Marks) 107. Solve : z ( p q 1) C 108. Find the integral surface of the equation ( y) y p ( y ) q ( y ) z passing through the curve z a, y Apply Charpit s method to find the complete integral of z p ay p q 110. z z Solve : cosm cosny 111. Find a surface passing through the lines z 0 and z 1 y 0satisfying z z z In the contet of a partial differential equation of the first order in there independent variables, define and illustrate the terms: (i) The complete integral (ii) The singular integral 11. w w w Find the general integral of ( y z w) ( z w) ( y w) y z z 114. Obtain the differential equation of the surfaces which are the envelopes of a one-parameter family of planes Eplain in detail the Charpit s method of solving the nonlinear partial differential equation z z f, y, z,, 0 y 116. z z z Solve z 1 1 y 117. Solve ( D 7D D 6 D ) z sin( y) e y y Find the differential equation of the family of all cones with verte at (,,1) 119. Find the integral surface of y y, z 1 z z which passes through the hyperbola p y q z 0, p, q Reputed Institute for IAS, IFoS Eams Page 10

11 10. Obtain a Complete Solution of pq m n l y z 11. Use the Charpit s method to solve16p z 9q z 4z 4 0. Interpret geometrically the complete solution and mention the singular solution. 1. Solve ( D DD' D' ) z y, by epanding the particular integral in ascending powers of D, as well as in ascending powers of D '. 1. Find a surface satisfying ( D DD') z 0 and touching the elliptic paraboloid z 4 y along its section by the plane y Find the surface whose tangent planes cut off an intercept of constant length R from the ais of z. 15. Solve ( y ) p ( y y) q ( y ) z 16. Find the integral surface of the partial differential equation ( y) p ( y z) q z through the circle z 1, y Using Charpit s method find the complete integral of z p qy pq Solve r s q z y 19. Find the general solution of r y t p yq log Solve: ( y z yz z y) p ( y z yz z y) q ( y z yz z y) 11. Find the complete integral of ( y )( qy p) ( p q) 1. Use Charpit s method to solve p qy z 1 pq 1. Find the surface passing through the parabolas z 0, y 4 a; z 1, y 4a and satisfying the differential equation r p Solve : r s 6t ycos u z z 15. Solve: z cos( y) e y Reputed Institute for IAS, IFoS Eams Page 11

Partial Differential Equations

++++++++++ 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