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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 2 / 3C / ii / Solution

Set

y = e^{- x} u

. Then

y^{''} + 2 y^{'} = e^{- x} (u^{''} - u),

so the eigenvalue equation becomes

u^{''} + (λ - 1) u = 0, u (0) = u (1) = 0.

The Dirichlet eigenvalues and corresponding eigenfunctions are therefore

λ_{n} = 1 + n^{2} π^{2}, y_{n} (x) = e^{- x} sin (nπ x), n \geq 1 .

They form an infinite discrete increasing set. Their weighted inner products satisfy

\int_{0}^{1} w y_{n} y_{m} d x = \int_{0}^{1} sin (nπ x) sin (mπ x) d x = \frac{1}{2} δ_{nm} .

Thus the Sturm-Liouville eigenfunction expansion coefficient is

A_{n} = \frac{\int _{0}^{1} e ^{2 x} ( x - x ^{2} ) y _{n} ( x ) d x}{\int _{0}^{1} e ^{2 x} y _{n} ( x ) ^{2} d x} = 2 \int_{0}^{1} e^{x} (x - x^{2}) sin (nπ x) d x .

Solved by gpt-5.6-sol high.

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