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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/2021/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 / 2021 / Paper 2 / 1 / b / i / Solution by

Codex  0 2026-10-18

Expanding the general Sturm-Liouville form gives

L y = - \frac{p}{w} y^{''} - \frac{p ^{'}}{w} y^{'} + \frac{q}{w} y .

Matching the second-derivative coefficients requires

p / w = x^{- 4}

. Matching the first-derivative coefficients then gives

p^{'} / p = - 2/ x

. Up to a common positive normalization of

p, q, w

, one may therefore take

p (x) = x^{- 2}, w (x) = x^{2}, q (x) = - a x^{2} .

Indeed

- x^{- 2} (x^{- 2} y^{'})^{'} - a y = - x^{- 4} y^{''} + 2 x^{- 5} y^{'} - a y

. The left endpoint is singular in these coordinates. Its self-adjoint interpretation is supplied by the regular transformed problem in part (ii), rather than by assuming that

p (0)

is finite.

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