 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2004/g/g1/i/solution

Multiply the equation for

y_{i}

y_{j}

and the equation for

y_{j}

y_{i}

, then subtract. The

q

terms cancel and the derivative terms combine into a Wronskian boundary derivative:

\frac{d}{d x} [p (y_{j} y_{i}^{'} - y_{i} y_{j}^{'})] + (λ_{i} - λ_{j}) w y_{i} y_{j} = 0.

Integrating over the interval eliminates the boundary term because both eigenfunctions vanish at both endpoints. Since their eigenvalues differ,

\int_{0}^{1} w (x) y_{i} (x) y_{j} (x) d x = 0.

This is orthogonality in a weighted inner product in the regular Sturm-Liouville setting; the endpoint values must justify the integration by parts.

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