You could try using the Laplace Transform in the time domain, which would give you a second-order ODE in the x domain. You then solve that using the usual methods, at which time the fun begins: computing the inverse Laplace Transform. I have no idea how difficult it would be to compute the inverse LT. The one time I did this, it was the diffusion equation (essentially the heat equation). The inverse LT was not pretty. I had to go back to its complex line integral definition, and compute residues and such. But it might be easier for you here. I don't know.