I am a bit confused over how to apply Duhamel's Principle (
en.wikipedia.org/wiki/Duhamel%27s_principle) when the forcing term depends on $t$, if it is in fact possible. In the first equations above the Wiki "Contents," it looks like the forcing term loses its dependence on $t$ when it gets moved to the initial condition, but in the "General considerations" section, it looks like it becomes a function of both $x$ and $s$.