I know that a solution to a pde I am interested is in the form: $ u(x,y)=(x-y)^{5}\frac{\partial^{4}}{\partial x^{2} \partial y^{2}}\left(\frac{F(x)-G(y)}{x-y}\right) $ where $F$ and $G$ are arbitrary functions to be determined.
For my case I know $u(x,y_{0})$ and $u(x_{0},y)$ explicitly. What is the best way to compute $F$ and $G$ in this case?