Solve the non-homogeneous wave equation
$u_{tt} = u_{xx}$
$u(0, t) = te^{-t}$ $u(x, 0) = 0$
$u(10, t) = 0$ $u_t(x, 0) = 0$
(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 $t = \{1, 2, 3, 4, 5\}$.