Start with the two linearly independent solutions to the spatial equation from part (c). Create a linear combination and impose the boundary condition at z = 0 to reduce your two linearly independent solutions to one. Then impose the boundary condition at x = L. You should only be able to satisfy this boundary condition for certain discrete values of k. What are these values of k (index them by an integer ne Z)? Write down a full basis of solutions un(r, t) with our two side boundary conditions imposed! Show that the solutions un(x, t) and u_(r,t) are linearly dependent and thus we only need to consider n = 1, 2,.... Hint (highlight to reveal): [The hi Inspired by Kreyszig, Section 12.6. 2