differential equations – Second order inhomogeneous PDE

I’m trying to get an exact solution to this second order inhomogeneous PDE:

$$
frac{partial^2}{partial{x}^2} y(x, z) – frac{partial^2}{partial z^2} y(x, z)=k^2y(x, z)-frac{1}{3}e^{4(x-2z)}y(x, z)
$$

where $k^2$ is a constant. No boundary conditions.
Any ideas? I tried with $y(x-cz)$ and with variable separation but it isn’t the right way.

Yesterday I shared the same question on Mathematics, maybe I’m luckier here