Suppose we have a configuration like the figure below.
I'm asked to compute the potential $\phi (x,y) $ inside the trap.
I'm attempting this by solving the Laplace equation $\nabla^2 \phi = 0$ because there is no charge in the interior. I'm doing this change of variables:
$u= xy$ ; $v= y$
And I'm impossing this boundary conditions:
$\phi (u=1,y) = 5$ and $\phi (u=-1,y) = -5$
However, I'm not able to solve the resulting PDE by separation of variables.
Doing some try and error with the boundary conditions, it seems that $\phi(x,y)=5xy$ solves the problem. But that's no rigorous at all.
I want to understand a mathematical rigorous way to get this same potential. In general, I'm getting problems trying to solve Laplace equation in non spherical, cylindrical nor plane symmetry.