(**Edit**: Reposted at Math.SE )

Psi_xx - Psi_tt - 4Psi = exp(exp(3it))*dirac_delta(x) DE valid for all x,t (no boundary conditions specified). Solve for Psi. If the DE is singular, then nontrivial solutions are okay.

-I solved the homogeneous portion, Psi_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_homogeneous like x*sin((4^0.5)*t) and x*cos((4^0.5)*t).

-With the guess method for Psi_particular, I have no idea what to guess on a general form of exp(exp(3it))*dirac_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_homogeneous and then looking for Psi_particular using the guess method...

-I'm curious if there is a general set of IC/BCs that I should be assuming as well?