It is about advanced mathematical methods

MATH 3333 (2020-Sep) 4 Question 2 (33 marks) This question concerns the differential equation 61:10: B'u(x,y) I 5 61' I (K+2) at2y)+(K1) 0' 1') +QK+$ + u(x, y) = 0 , where K is a real constant. (a) Explain that the given differential equation can never be elliptic. [4] (b) State the value of K for which the equation is [6] (i) hyperbolic; (ii) parabolic. (c) Consider the case K = 1. Use the method of separation of variables, u(x, y) = X (x)Y (y) , to show that the functions X (x) and Y0) satisfy the partial differential equations X "(x) + AX (x) = 0, Y'(y)+(13A)Y(y)=o, for any positive constant 2t. Hence determine the general solution u(x, y) satisfying the given boundary conditions \"(0.3%) = \"(1,y) = 0 (y 2 0). u(x, 0) = sin(trx) si11(31rx) (0 S x S l). l''l