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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/2015/paper-2/1/b/i/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 / 2015 / Paper 2 / 1 / b / i / Solution by

Codex  0 2026-10-18

To obtain Sturm-Liouville form, multiply the equation by

(1 - x^{2})^{- 1/2}

. The derivative identity

\frac{d}{d x} [1 - x^{2} y_{n}^{'}] = 1 - x^{2} y_{n}^{''} - \frac{x}{1 - x ^{2}} y_{n}^{'}

gives

- \frac{d}{d x} [1 - x^{2} y_{n}^{'}] = n^{2} \frac{y _{n}}{1 - x ^{2}}, p (x) = 1 - x^{2}, q (x) = 0, w (x) = \frac{1}{1 - x ^{2}} .

The weight determines the weighted inner product on

(- 1, 1)

. It is integrable but singular at the endpoints, where

p

also vanishes. Thus this is a singular-endpoint Sturm-Liouville problem, not a regular problem with strictly positive endpoint values of

p

. The transformation in part (ii) specifies its admissible Dirichlet realization; these are the Dirichlet eigenfunctions of the Chebyshev differential operator.

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