QUESTION FOUR (a) Express the function f(x) = (1, when |x| 1 10, when |x|>1 as a Fourier Integral, hence evaluate sin cos x d (i) Hyperbolic (ii) Parabolic (iii) Elliptic dx (b) Solve the differential equation - y = et, + x = sin t given that x(o) = 1, and y(0) = dt dt 0 by Laplace transforms. [8 Marks] QUESTION FIVE (a) Given the pde (1 + y)Uxx + 2(1 x)Uxy + (1+y)Uyy = U. Determine the values of x and y for which the equation is; [7 Marks] (b) Solve the pde Uxx + Uxy - 6Uyy = cos(2x + y) by the D. Operator. (c) Evaluate te-3t sin t dt by Laplace transforms. [2 marks] [2 marks] [2 marks] [6 Marks] [3 Marks]