 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/1/i/solution

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

. Since

\frac{d}{d x} [1 - x^{2} y^{'}] = 1 - x^{2} y^{''} - \frac{x}{1 - x ^{2}} y^{'},

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

⟨ u, v ⟩_{w} = \int_{- 1}^{1} u^{*} v d x / 1 - x^{2}

. These endpoints are singular:

p

vanishes and

w

diverges, though

w

is integrable. The required boundedness of

y, y^{'}

makes the boundary factors

p y^{'}

vanish, which is the fact needed in the integration identities below.

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