(0). Solve 3x 2+ fo* y(t) sin(x — © dt = y(x) (4 marks)

5 (a). Solve the integral equation y(x) = 1 + \ Jn xty (t)dtpy the method of successive

approximations (4 marks)
(b). Reduce to the canonical form x 2 — 2xyuxy + Yuyy=“™ (5 marks)
6 (a). Using Laplace transforms solve the IBVP
uxx — Futt —COS ot,0S x < 00,0{ < 7, ०८00 -- 0,
u is bounded as x tends to 00, ut(x, 0) = u(x, O) =0 ട്ട
(0). When the second order partial differential equation
R.uxx + ऽ. 1 + 1. 1+ u, uX, uy) =O
is said to be (i) Elliptic (11) Parabolic and (111) hyperbolic (3 marks)
Part C
7 (a).Prove that B (m, n) = (6 marks)
r(m+n)
(b). Deduce that r(h) = from 3 (m, 1) ಎಷ (2 marks)
(c). Prove hat 1 (2८) ಎ. sinx (4 marks)
Ttx
8 (a).Prove that cos(xsin9) = Jo(x) 2cos29 + J4(x). 2cos40 + (4 marks)

(b). Express the polynomial x3 — 5x2 + x + 2 in terms of Legendre's polynomials
(3 marks)

(c).State and prove the orthogonality property of Legendre polynomials (5 marks)
9. Using Crank Nicholson method solve प = 16,0 Ogiven u(x, 0) =O,
u(x, t) = 0, u(1,t) = sot. Compute प for 2 steps in t direction taking
(12 marks)