Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 2 / a

Take a nonzero separated solution $\phi =X(x)Y(y)$. Dividing by $XY$ separates the Laplacian eigenfunction equation into $X^{\prime \prime }=-\mu X$ and $Y^{\prime \prime }=-(\lambda -\mu )Y$. The zero-endpoint Dirichlet boundary conditions admit no nonzero linear or hyperbolic factors, so both separation constants must be positive squares. HenceAll separated nonzero solutions have this form. At a repeated eigenvalue, linear combinations of the corresponding modes are also solutions, though they need not remain separable.