Assuming that $F(x,y) = f(x) + g(y)$ what is the solution of the following partial differential equation? \begin{equation} \left(\dfrac{\partial F}{\partial x}\right)\left(\dfrac{\partial F}{\partial y}\right) + xy = C \end{equation} Here, $C$ is a constant.
Is there a general scheme to solve such PDEs? It would be appreciated if such scheme is provided, since I need to solve more such PDEs. For instance, how to solve the following PDE? \begin{equation} \left(\dfrac{\partial F}{\partial x}\right)\left(\dfrac{\partial F}{\partial y}\right) + xy + \gamma x\dfrac{\partial F}{\partial x} = C' \end{equation}
