b. Solve
YO) = 2 + fo” y(t). cos — t) dt, y(O) = 1 (5 marks)

a. By the method of successive approximation solve the integral equation

YO) = 1 + ۸ fo*xt. y(t)dt (4 marks)

b. Reduce to the canonical form the PDE uxx + x2u (5 marks )

Solve the IBVP using Laplace transform technique
1
UXX ய்‌ = —dautt — coswt,O 5 x >
00, u(0,t) = 0, u Ot
२६ (2, 0) = u(x,0) = 05 boundedast ௦0,
(9 marks )

PART C

(a) Prove that =—Cosx (3 marks )
(b) State and prove the orthogonality property of the Bessel function (5 marks)
O Prove that (4 marks )
8. (a) Prove that ॥0+2.12+214+................ =i (4 marks (
(b) State and prove the Rodrigue formula
(5 marks )
O Evaluate sin99cos59 de (3 marks )

9. solve PDE

४८५८ + ‏ں‎ + 10 )×2+ 2+ 10( = 0
over the square with sidesx= = = 3 and ५ = 3 with u = 0 on the boundary and
mesh length!. (12 marks )