Problem:
$ \Psi_{xx} - \Psi_{tt} - \Psi = \exp(3t) \cdot \delta(x)$
No boundary conditions specified.
I solved the homogeneous portion, $\Psi_\mathrm{homogeneous}$, of this equation via separation of variables but my solution is just for some random case of $k^2$ where: F''/F = G''/G + 4 = k^2. I chose the case where $k^2 = 0$ which gave me solutions for $\Psi_\mathrm{homogeneous}$ like $x\sin(2t)$ and $x\cos((2t)$.
With the guess method for $\Psi_\mathrm{particular}$, I have no idea what to guess on a general form of $\exp(3t)\delta(x)$ to plug back in to the PDE.
I have read about Green's Functions but, man, I'm having a hard time understanding the guides I have seen because they skip so many of the intermediate steps. I understand that these Green's Functions can provide a general solution and that seems like what I'm looking for. I likely spent a lot of time for nothing on my first attempt with separation of variables for $\Psi_\mathrm{homogeneous}$ and then looking for $\Psi_\mathrm{particular}$ using the guess method...
I'm curious if there is a general set of IC/BCs that I should be assuming as well?