Solution
= Solution
For $k>0$, the two ordinary differential equations have solutions
$$
\boxed{h(x)=A\cosh kx+B\sinh kx,\qquad g(x)=C\cos kx+D\sin kx.}
$$
Their 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.