Question 2. (16 marks) Consider the inhomogeneous one-dimensional heat equation du Fu = D + (1 - x ) cost, 00 (1) at with homogeneous boundary conditions axu(0, t) = 0, u (1, t) = 0, t >0. and the initial condition u(x, 0) = 0. (a) Show that separation does not work for (1). (1 marks) (b) Expand the t-independent factor of the inhomogeneous term in (1), i.e. (1 - x-), 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) (d) Solve the first-oder ODEs of (c). (5 marks) (e) Impose the initial condition to arrive at the (formal) solution for u (x, t). (2 marks)