Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 2 / b / i / Solution

For $\Phi =X(x)Y(y)$, separation of variables gives $X^{\prime \prime }/X+Y^{\prime \prime }/Y=-(m^2+p^2)$. With the standard separation constants $m^2$ and $p^2$, the zero-endpoint solutions are sines. Thus, provided $mp\ne 0$,is nonzero, has zero value on all four sides, and satisfies $\Delta \Phi =-(m^2+p^2)\Phi$.

There is a missing qualification in the printed request for arbitrary integer $m,p$. The Dirichlet and Neumann spectra of a square show that a nonzero Dirichlet solution exists precisely whenWhen one of $m,p$ is zero, the displayed sine product vanishes identically, and a nonzero solution need not exist: for example $(m,p)=(0,0)$ or $(0,1)$ admits none. On the other hand $(0,5)$ does admit $\sin (3x)\sin (4y)$. Thus the natural intended case is two nonzero integer indices, with the stated spectral criterion covering the exceptional cases.