An electrical signal described by the function u(x,t) is transmitted in the x direction. The signal satisfies the wave equation,

subject to the conditions

for x > 0,

for t > 0, where q(x) is a known function.

Find the solution for u(x,t).

So far I have done...

The product solution is:

Plugging into boundary conditions gives:

Plugging the product solution into the DE and separating gives:

From separting variables we get 2 ODES:

And then I'm really really stuck about how to do the rest, I think I need to get some eigenfunctions and eigenvalues, but then don't know how to get the solution u(x,t) from there...

Any help would be so appreciated, I need to know how to do loads fo these questions for my exam but I'm really struggling

Thanks in advance!