I am tryng to solve $$\nabla ^2 \phi = -\rho(\vec x)$$ Where $\phi(0) =0$ and $\phi(L)=\phi_0$ and $\rho$ is an arbitrary functionAccording to wolfram we can write G in terms of spherical harmonics, which then leaves me wondering how I can have the boundary conditions met from $$\phi = \int G\rho d^3x$$ I know I can add a solution to laplaces equation (maybe $\phi = \phi_0 x/L$) but other that that I am not sure.
The rationale is that I have an electrostatics problem of an arbitrary charge distribution in between two conducting plates, one at 0V and the other at Vo, which reduced to this BVP.