 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2017/6/a/iii/solution

Multiply the equation for

y_{n}

y_{m}

and that for

y_{m}

y_{n}

, then subtract. Integrating the resulting identity yields

[p (y_{m} y_{n}^{'} - y_{n} y_{m}^{'})]_{x_{0}}^{x_{1}} + (λ_{n} - λ_{m}) \int_{x_{0}}^{x_{1}} w y_{n} y_{m} d x = 0.

All endpoint terms vanish because both functions satisfy the zero Dirichlet boundary conditions. Since the multipliers differ,

\int_{x_{0}}^{x_{1}} w (x) y_{n} (x) y_{m} (x) d x = 0.

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