5. Solve the non-homogeneous heat equation by using the method of separation of variables
$u_t - ku_{xx} = x$, $0 < x < L$, $0 < t$, $u = u(x, t)$
subject to inhomogeneous boundary conditions, where $k$ is a constant,
$u(0, t) = 1$, $u_x(L, t) = 1$
and initial condition
$u(x, 0) = x(L - x)$