Let \(L>0\) and \(R=(-L, L)\times (-1, 1)\). All solutions \(u\in C^2(R)\cap C(\bar{R})\) of the Laplace’s equation of the form \(u(x,y)=v(x)w(y)\) with \(u(x,y)=0\) for all \((x,y)\in (-L,L)\times \{-1,1\}\) are given by

Proof

After applying the Laplace operator on \(u(x,y)=v(x)w(y)=0\) we obtain

\[ \frac{v''(x)}{v(x)} = - \frac{w''(y)}{w(y)}=K \]

for some \(K\in \mathbb{R}\).