= Solution
Take a nonzero separated solution $\varphi=X(x)Y(y)$. Dividing by $XY$ separates the <Laplacian eigenfunction> equation into $X''=-\mu X$ and $Y''=-(\lambda-\mu)Y$. The zero-endpoint <Dirichlet boundary conditions> admit no nonzero linear or hyperbolic factors, so both separation constants must be positive squares. Hence
$$
\boxed{\varphi_{mn}(x,y)=C\sin\frac{m\pi x}{\ell}\sin\frac{n\pi y}{\ell},\qquad
\lambda_{mn}=\frac{\pi^2(m^2+n^2)}{\ell^2},\quad m,n\geq1.}
$$
All 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.
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