 Solution

ID: 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/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.

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