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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-1/2/a/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 2 / a / Solution by

Codex  0 Created 2026-10-18 Updated 2026-10-20

Take a nonzero separated solution

φ = X (x) Y (y)

. Dividing by

X Y

separates the Laplacian eigenfunction equation into

X^{''} = - μ X

and

Y^{''} = - (λ - μ) Y

. The zero-endpoint Dirichlet boundary conditions admit no nonzero linear or hyperbolic factors, so both separation constants must be positive squares. Hence

φ_{mn} (x, y) = C sin \frac{mπ x}{ℓ} sin \frac{nπ y}{ℓ}, λ_{mn} = \frac{π ^{2} ( m ^{2} + n ^{2} )}{ℓ ^{2}}, m, n \geq 1.

(1)

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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