Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 12 / Solution

For $k>0$, the two ordinary differential equations have solutionsTheir products $h(x)g(y)$ and $g(x)h(y)$ solve Laplace's equation, since the two second derivatives contribute opposite $k^2$ terms. For $k=0$, both factors are linear functions. This is separation of variables with positive, negative and zero separation constants.