Question 2. (16 marks) Consider the inhomogeneous one-dimensional heat equation au au = D + (1 - x2) cost, at 00 (1) with homogeneous boundary conditions axu(0, t) = 0, u( 1, t) = 0, t > 0. and the initial condition u(x, 0) = 0. (b) Expand the t-independent factor of the inhomogeneous term in (1), i.e. (1 - x2), in terms of eigenfunctions of the boundary value problem. (4 marks) (c) Expand u(x, t) in terms of the eigenfunctions of the boundary value problem with t-dependent coefficients. Using the result of (b), derive first-oder ODEs for these coefficients. (4 marks)