QUESTION 6 a) Consider the following wave equation 1 Uu = - 00 < a < 00, with initial conditions u(x,0) = sin Tz, %3D u(x, 0) = 0. Use d'Alembert method to determine the solution of the above problem. b) Consider the following heat equation du Pu 0 0, with boundary conditions u(0, t) = 0, u(1, t) = 0, t>0, and initial condition u(x,0) = f(x), 0 < < 1. i. Use separation of variables to show that the heat equation can be reduced to X" kX = 0, X'(0) = 0, X'(1) = 0, T- 4KT = 0, where k is a separation constant. ii. Hence, find the solution of the heat cquation if f(x) = x(1 x), 0