Program : M.A./M.Sc. (Mathematics) M.A./M.Sc. (Previous) Paper Code:MT-03 Differential Equations, Calculus of Variations & Special Functions Section C (Long Answers Questions) 1. Solve 2x cos y 2x sin y + x cos y sin y = Log x A. MT-03, P. 4 2. Solve: t (1 log y) + (1 + log y) = 0 n A. MT-03, P. 17, 18 3. Solve : = e 2 + 4 = 0 a = 1 + A. MT-03, P. 16, 20 4. Solve : / = cos x y sin x + y + x = 0 A. MT-03, P. 11, 15 5. Solve : (y + z + x ) dx 2xy dy 2xz dx = 0 (zy + z )dx xz dy + xy dx = 0 A. MT-03, P. 30, 33 6. Solve : (2xz yz)dx + (2yz xz)dy (x xy + y )dz = 0 3x dx + 3y dy (x + y + e )dz = 0 A. MT-03, P. 34, 37 7. Solve : yz (y + z)dx + zx (x z)dy + xy (x + y)dx = 0 y z (x cos x sin x) dx + x z (y cos y sin y) dy + xy (y sin x + x sin y + xy cos z) dx = 0 A. MT-03, P. 35, 38
8. Solve xz dx zdy + 2 ydx = 0 A. MT-03, P.36 9. Solve : dx dy tan ax = 0 (2x + y + 2xz) dx + 2xydy + x dz = dt A. MT-03, P. 31, 42 10. Solve r + (a + b)s + abt = xy by monge s method A. MT-03, P. 52 11. Solve : t r sec y = 2 q tan y (rt s ) s (sin x + sin y) = sin x siny A. MT-03, P. 56, 61 12. Solve : t + s + q = 0 z (1 + q )r 2pqz s (1 + p )t + z (rt + s ) + 1 + p q = 0 A. MT-03, P. 47, 62 13. Solve : 2s (rt s ) = 1 x r + 2xys + y t = 0 A. MT-03, P. 59, 54 14. Solve : q r 2pqs + p t = 0 3r + 4s + t 6 (rt s ) = 1 A. MT-03, P. 55, 58 15. Find the characteristics of : y r x t = 0 x r + 2x sy + y t = 0 A. MT-03, P. 73, 74 16. Solve the following P.D.E. by the method of separation of variables. = 2 + u given that u(x, 0) = 6e 2 + = 0 A. MT-03, P. 75, 76 17. (a) classify the equations : + 2 + = 2 + 2 + + = (b) Solve by the method of separation of variables the PDE 4 + = 3u given that u = 3e e when t = 0 A. MT-03, P. 68, 77 18. Reduce the equation xyr (x y )s xyt + by qx = 2(x y )
to canonical form and hence solve it A. MT-03, P. 97 19. Reduce the following PDE to canonical form. (n 1) y = n y + 2 + = 0 A. MT-03, P.93, 94 20. Find eigen values and eigen functions for the following boundary problems y + λy = 0 ; y (a) = 0 and y(b) = 0 ; 0 < a < b ; b are real arbitrary constants. y 4y + (4 9λ)y = 0 ; y(0) = 0, y(a) = 0 ; where a is a positive real constant. A. MT-03, P. 105, 108 21. Prove that : The eigen values of strom-liouville system are real. To every eigen value of a strom-liouville system there corresponds only one linearly independent eigen function. A. MT-03, P. 113, 115 22. Compute the eigen values and eigen functions for boundary value problem y + 2y + (1 λ)y = 0 ; y(0) = 0, y(1) = 0. Also prove that the set of eigen functions for the given problem is an orthogonal set. A. MT-03, P. 117 23. Find the shape of the curve on which a bead is sliding from rest and accelerated by gravity will ship (without friction in least time from one point to another. A. MT-03, P. 129 24. (a) Find the extremum of the function: F[y(x)] = (1 + y ) / dx x (b) Show that the curve through (1, 0) and (2, 1) which minimize : ( ) / dx is a circle. A. MT-03, P. 133 25. Test for extremum of the functional. (a) F[y(x)] = 1 + y dx, y(0) = 0, y(1) = 2 / (b) F[y(x)] = (y y ) dx, y(0) = 0, y(π/2) = 1 A. MT-03, P. 128, 135 26. Test for an extremum of the functional. (a) F[y(x)] = (x y + x y ) dx, y(0) = 0, y(1) = 1 (b) F[y(x)] = (y + x ) dx, y(0) = 0, y(1) = 2 A. MT-03, P.127, 129
27. (a) If y(x) is a curve in internal [a, b] which is a twice differentiable and satisfying the condition y(a) = y and y(b) = y and minimizes the functional F[y(x)] = f(x, y, y ) dx, where y = Then the following differential equation must be satisfied f y d f dx y = 0 (c) Obtain the Euler-Lagrange equation for the extrremals functional [y yy + y ] dx A. MT-03, P. 124, 135 28. Determine the curve of prescribed length 2 l which joins the points (-a, b) and (a, b) and has its centre of gravity as low as possible. A. MY-03, P. 152 29. Find the shape assumed by a uniform rope when suspended by its end from two points at equal heights. A. MT-03, P. 151 30. Find enternal of the functional: / (a) I = 1 + y dc, y(0) = 0, t (0) = y(1) = y (1) = 1 (b) I = y y + x dx, y(0) = y (0) = 0 = y(π/2), y (π/ 2) = 1 A. MT-03, P. 147-148 31. (a) Find the closed convex curve of length L that encloses greatest possible area. (b) Find the external equation for the following functional I[z(x, x ] = z + z dx x x dx A. MT-03, P. 150, 148 32. Solve the Gauss hyper geometric equation. x(1 x) d y dy + [γ (1 + α + β)x] dx dx αβy = 0 In series in the neighborhood of the regular singular point x = 0 and x = 1 A. MT-03, P. 166 33. Solve in series: (a) (2 x )y + 2xy 2y = 0 (b) 2x y xy + (1 x )y = 0 A. MT-03, P. 161, 164 34. Solve in series : (a) (1 x ) y 2xy + n(n + 1)y = 0 (b) x(1 x)y + (1 5x)y 4y = 0
A. MT-03, P. 162, 170 35. Solve in seriesl (a) x y + xy + (x 1)y = 0 (b) x y + (x + x )y + (x 9)y = 0 A. MT-03, P. 173,176 36. Show that if b > 0 / Where ξ = 2f (a, b, 2b; z) = 2 1 z 2 2 B(b, b) (sin ) [(1 + ξ cos) + (1 ξ cos) ]d Deduce that 2f (a, b; 2b ; z) = 2 1 z 2f,, + ; b + ; ξ A. MT-03, P. 191 37. Prove that L (a) 2f,, + ; ; z = [(1 z) + (1 + z) ] (b) 2f ( n, a + n ; c ; 1) = () () () A. MT-03, P. 190 38. Show that of x, sin hx = n sin x 2f 1 2 + 1 2 n, 1 2 1 2 n; 3 2 ; sin n and cos n x = 2f, ; ; sin x A. MT-03, P. 193 39. Prove that : (a) B(λ), ( λ)2f (a, b ; c, z) = t (1 t) (a, b; ; zt)dt Where z < 1, λ > 0, c λ > 0 (b) 2f (a, b; c; z) = Where c>b>0 hence prove that 2f (1,2; 3; z) = log e(1 z) / A. MT-03, P. 191, 196 40. Prove that : (,) u (1 u) (1 zu) du (a) 2f, + ; ; z = [(1 z) + (1 + z) ] (b) lim 2f Ґ() (a, b; c; z) = () () () 2 ; z) A. MT-03, P. 190, 195 41. Show that: 2f (a + n + 1, b + n + 1; n +
(a) 2f [ n, a + n; c; 1) = () () / () (b) 2f (, ; 1; k ) = A. MT-03, P. 190, 196 42. Show that : (a) Ґ(a) Ґ(b) 2f a, b; ; z = e cos h 2(uv) z u v dudv Provided Re (a) > 0 and Re (b) > 0 (b) x (t x) [1 x (t x) ] = πt 2f, A. MT-03, P. 194, 198 43. If z < 1 and < 1 then (a) 2f (a, b; c; z) = (1 z) 2f (a, c b; c; (b) 2f (a, b; c; z) = (1 z) 2f (c a, b; c; A. MT-03, P. 202 44. Show that : (a) [2f (a, b; c ; x] = ) ) ; 1; 2f (a + 1, b + 1 ; c + 1 ; x) (b) [2f (a, b; c ; x] = () () 2f () (a + n, b + n ; c + n ; x) A. MT-03, P. 204 45. Show that by the principle of mathematical induction: d dx [2f (a, b; c; x)] = (a) (b) 2f (c) (a + n, b + n ; c + n ; x) A. MT-03, P. 204 46. Show that : (a) 1f (a; c; n) = 1f (a + 1 ; c + 1 ; x) (b) [1f (a; c ; x] = () 1f () (a + n, ; c + n ; x) A. MT-03, P. 211 47. If z < 1 and < 1 then: 2f (a, b ; c ; z) = (1 z) 2f (c a, b ; c; ) 2f (a, b ; c ; z) = (1 z) 2f (c a, c b ; c; z) A. MT-03, P. 202 48. If z < 1 and < 1 then: 2f (a, b ; c ; z) = (1 z) 2f (a, c a, ; c; ) 2f (a, b ; c ; z) = (1 z) 2f (c a, c b ; c; z) A. MT-03, P. 202 49. If z < 1 and < 1 then: 2f (a, b ; c ; z) = (1 z) 2f (a, c a, b ; c; and deduce that: ) 2f a, 1 a ; c ; = 2 2f (a, c + a ; c; 1)
2f a, 1 a ; c ; = Ґ Ґ( ) Ґ Ґ( ) A. MT-03, P. 202, 203 50. Show that by the principle of mathematical induction. d dx [1f (a; c; x)] = (a) 1f (c) (a + n; c + n; x) A. MT-03, P.11 51. If m is a positive integer, show that: 2f ( m, a + m; c; x) = x (1 x) Ґ(c) dm Ґ(m + c) dx m x c+m 1 (1 x) a c+m and deduce that: 2f m, a + m; a 2, 1 2 ; 1 2 1 (μ 2 μ = 1) Ґ( a 2 + 1 2 ) d 2 m Ґ( 1 2 + a 2 + m) A. MT-03, P. 212 52. Solve : (1 x )y 2xy + n(n + 1)y = 0 where n is positive integer. A. MT-03, P. 219 53. Prove that : P (x)p (x)dx = 0 if m n [P (x)] dx = A. MT-03, P. 225 54. Prove that : if m = n (n + 1)P (x) = (2n + 1)xP (x) np (x), n 1 np (x) = xp (x) P (x) A. MT-03, P. 226 55. Prove that: x P P dx = xp P dx = A. MT-03, P. 233, 234 56. Show that : () ()()() P (x) = x ± (x 1) cos θ dθ P (x) = x ± (x 1) cos () d A. MT-03, P. 232, 234 57. Show that : Q Q = (2n + 1)Q nq + (n + 1)Q = (2n + 1)x Q A. MT-03, P. 236, 237 58. Show that : Q Q = (2n + 1)Q (2n + 1)(1 x )Q = n(n + 1) (Q Q ) A. MT-03, P. 236, 238 59. Show that : m dx m μ 2 1 m+ a 2 1 2
nq +(n + 1)Q = (2n + 1)x Q (2n + 1)x Q = (n + 1) Q + nq A. MT-03, P. 237, 238 60. Prove that (x 1) Q P P Q = C and deduce that : ( ) Q (n) = log A. MT-03, P. 240 61. Show that : x J (x) = n J (x) x J (x) x J (x) = x J (x) x J (x) A. MT-03, P. 252, 253 62. Show that : x J (x) = n J (x) x J (x) [x J (x)] = x J (x) A. MT-03, P. 252, 253 63. Show that: x J (x) = n J (x) n J (x) [x J (x)] = x J (x) A. MT-03, P. 253, 254 64. By using generating function or Bessel function show that: cos(x sin θ) = J + 2J cos θ + 2J cos 4θ +. Sin(x sin θ) = 2J sin θ + 2 3 sin 3θ +. (iii) cos x = J 2J + 2J.. (iv) sin x = 2J 2J + 2J. A. MT-03, P. 255 65. If λ and λ are the roots of equation J (λ ) = 0 then 0 if i j J (λ, x)j λ, xdx = (λ, a) if i = j A. MT-03, P. 260 66. Prove that : a 2 J π n + J (x) = exp(ixt)(x t ) dt, (n > ) (a x ) J (kx)dx = J (ak) J (ak) A. MT-03, P. 264, 266 67. Prove that: [xj (x)j (x)] = x[j (x) J (x)] (x)j (x) = 2 J (x) J (x) A. MT-03, P. 255, 256 68. Show that : 2xH (x) = 2n H (x) + H (x) H (x) = 2x H (x) (n 1) (iii) H (x) = 2x H (x) H (x) (iv) H (x) 2x H (x) + 2n H (x) = 0 A. MT-03, P. 273
69. Prove that : If m n [H (x)] = H (x) H (x) = ( 1) e (e ) A. MT-03, P. 275, 276 70. Prove that : 0, if m n e H (x)h (x)dx = π 2 n if m = n H (x) = 2 exp x A. MT-03, P. 278, 279 71. Prove that : P (x) = e t H (xt)dt () = exp(2xt t )Hs (x t) A. MT-03, P. 281, 282 72. Show that : (n + 1) n + 1 (x) = (2n + 1 x) n(x) n n 1 (x) x n (x) = n n(x) n n 1(x) A. Mt-03, P. 288, 289 73. Establish the generating functions : Ґ(1 + α)(xt) e J 2 xt = 1F () C ; 1 + α; = A. MT-03, P. 300, 301 74. Prove that : e (t)dt = 1 (x) (x) = A. MT-03, P. 292, 297 75. Show that: (x) + (x) = (x) (n + 1) A. MT-03, P. 295 76. Prove that : () (x)t () () (x)t = (2n + k + 1 x) (x) (n + k) (x) e 0 if m n (x) (x)dx = 1 if m = n () exp L (x)t A. MT-03, P. 290, 294 =