Use D'Alembert's solution to solve the following
subject to the initial conditions and .
Note that with and
With your change of variables
Integrating twice gives
Imposing the two boundary conditions gives
Differentiating the first and then solving for F' and G' gives
Finally, we have
but imposing the first BC (1) gives , so the final solution is
First the D'Alembert solution applies for
But your PDE has a nonhomogeneous term.
So we'll find a particlar solution - works. So let where now satisfies
and your IC's gives and