Solve the non-homogeneous diusion equation ut = 2uxx + 4x ux (0, t) = 0 u (1, t) = 1 u (x, 0) = x 2 (a) Separate the problem into boundary and interior parts. (b) Find all the eigenfunctions, and transform the interior PDE. (c) Solve the resulting ODEs and invert to write the solution as a series. (d) Plot a 5-term series solution for times t = {0.0, 0.1, 0.4, 0.9, 1.6, 2.5} . For a bonus point, create and upload a 51-frame movie spanning the time domain t ∈ [0, 2.5].