Nonhomogeneous wave equation (18 Marks) The method of eigenfunction expansions is often useful for nonhomogeneous problems re- lated to the wave equation or its generalisations. Consider the problem Urt = [p(x) uxlx-q(x)u+ F(x, t), I ux(0, t)-hu(0, t) = 0, ux(1, t) + h₂u(1, t) = 0, u(x,0) = f(x), u, (x,0) = g(x). 1.1 Derive the equations that X(x) satisfies if we assume u(x, t) = X(x)T(t). [5] 1.2 In order to solve the nonhomogeneous equation we can make use of an orthogonal (eigenfunction) expansion. Assume that the solution can be represented as an eigen- function series expansion and find expressions for the coefficients in your assumption as well as an expression for the nonhomogeneous term. [13]