QUESTION TWO Solve the following pde's by the operator method. (i) Uxxx - 2Uxxy = 2ex + 3xy (ii) Uxx + Uxy - 6Uyy = y cos x [8 Marks] [7 Marks] QUESTION THREE (a) Given the pde Uxx = U,U(x,0)=x (25-x. Solve the pde by the method of separation of variables. [6 Marks] (b) A string is stretched and fastened to two point < apart. Motion is started by displacing the spring from which it is released at a time t = 0. Find the displacement at any point at a distance x from one end at time t, given that the equation of vibration of the string is Utt = cUxx under the following conditions; [9 Marks] (i) U(0, t) = 0 (ii) U(0, x) = 0 (iii) (U(l, t) = 0, (iv) U(x, 0) = 10 sin "x