Use D'Alembert's solution to solve the following
subject to the initial conditions and .
Note that with and
With your change of variables
gives
Integrating twice gives
or
Imposing the two boundary conditions gives
.
Differentiating the first and then solving for F' and G' gives
.
Thus, .
Finally, we have
but imposing the first BC (1) gives , so the final solution is
I used it after making the change of variables. Let me derive my solution with the D'Alembert solution.
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
,
D'Almbert solution
and your IC's gives and
so
.