SCHWEITZER 1961 Problem 7

For the differential equation $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}= 2\frac{\partial^2 u}{\partial x \partial y}$ find all solutions of the form $u(x,y)=f(x)g(y)$ . (R. 14)