Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 1 / d

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- Solution d

Solution

Apply the identity from (c) to the two real eigenfunctions. The equations of their Sturm-Liouville problem give

0 = \int_{- L}^{L} (f_{n} L f_{m} - f_{m} L f_{n}) d x = (λ_{m} - λ_{n}) \int_{- L}^{L} w f_{n} f_{m} d x .

Since the eigenvalues differ,

\int_{- L}^{L} w (x) f_{n} (x) f_{m} (x) d x = 0.

This is orthogonality in the weighted inner product.

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