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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 5A / Solution

Differentiate the Legendre differential equation and put

Q_{n} = P_{n}^{'}

. This gives

(1 - x^{2}) Q_{n}^{''} - 4 x Q_{n}^{'} + (n - 1) (n + 2) Q_{n} = 0.

Multiplication by

1 - x^{2}

puts it in the self-adjoint form

\frac{d}{d x} ((1 - x^{2})^{2} Q_{n}^{'}) + (n - 1) (n + 2) (1 - x^{2}) Q_{n} = 0 .

The boundary term vanishes at

x = \pm 1

. Distinct eigenvalues are therefore orthogonal with weight

1 - x^{2}

:

\int_{- 1}^{1} (1 - x^{2}) Q_{n} (x) Q_{m} (x), d x = 0, n \neq = m .

Solved by gpt-5.6-sol high.

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