= Solution
Seek a separated nonzero solution $u=X(x)Y(y)$. Division by $XY$ on intervals where the factors are nonzero gives $X''/X+Y''/Y=-25$. Write $X''+\lambda X=0$ and $Y''+(25-\lambda)Y=0$, with both factors zero at their endpoints $0,\pi$.
For $X$, a nonpositive $\lambda$ gives only the zero solution under the two endpoint conditions: linear <functions> vanish identically when $\lambda=0$, and the hyperbolic solutions do likewise for $\lambda<0$. For $\lambda>0$, the endpoints force $X=A\sin(mx)$ with positive integer $m$ and $\lambda=m^2$. Applying the same argument to $Y$ gives $Y=B\sin(ny)$ and $25-m^2=n^2$, with positive integer $n$.
The only positive integer pairs satisfying $m^2+n^2=25$ are $(3,4)$ and $(4,3)$. By <superposition>,
$$
\boxed{u(x,y)=A\sin3x\sin4y+B\sin4x\sin3y,\qquad A,B\in\mathbb R.}
$$
To establish that this is the most general sufficiently regular solution, expand $u$ in the complete product <Fourier sine series> on the square. Substitution, or <integration> by parts against each <sine> mode, gives $(25-m^2-n^2)c_{mn}=0$ for every <coefficient>. Hence only the two displayed modes remain; all boundary terms vanish because of the given zero values. This is the eigenvalue-$25$ case of the <Dirichlet eigensystem of a rectangle>. The zero solution is included when $A=B=0$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1-square-eigenmodes.png]
{title=The two independent zero-boundary modes sharing Laplacian eigenvalue 25 on the square}
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