I'm trying to solve a non-homogeneous wave equation in cylindrical coordinates \begin{align} \nabla^2\mathrm A-\frac{\partial^2\mathrm A}{\partial t^2}=\mathrm J, \end{align} where A and J are cylindrically symmetric, i.e. they are functions of $s$ and $t$, (J is a separable function). The nabla operator can be written out explicitly, \begin{align} \frac{\partial^2\mathrm A}{\partial s^2}+\frac{1}{s}\frac{\partial\mathrm A}{\partial s}-\frac{\partial ^2\mathrm A}{\partial t^2}=\mathrm J. \end{align} If this were spherically symmetric, the $1/s$ would be a $2/s$ and I could solve this using the transformation $s$A$(s,t)$=B$(s,t)$ and d'Alembert's solution method, but I believe that doesn't work here. I also cannot find solution methods for non-homogeneous cylindrically symmetric wave equations online that don't involve separation of variables.
For a homework assignment I'm supposed to show that a certain solution satisfies this equation, (not to derive it, I would just like to), but it's complicated enough that I can't imagine someone guessed it, as well as not being separable. I could give more specifics on J and the solution, but I thought I'd keep it more general for the site.