Eigenvalues of the Laplacian on a square
What are the solutions to the equation uxx + uyy = λu on the unit square with the requirement that u(x, y) = 0 on the boundary? It’s easy to see that the functions u(x, y) = sin(mπx) sin(nπy) are solutions with λ = (m² + n²)π² for non-negative integers m
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