 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-2/1/iv/solution

y_{i}, y_{j}

, multiply their Sturm-Liouville problems crosswise and subtract. Integration gives

(λ_{i} - λ_{j}) \int_{1}^{R} x^{2} y_{i} y_{j} d x = [x (x - 1) (y_{i} y_{j}^{'} - y_{i}^{'} y_{j})]_{1}^{R} .

R

, both eigenfunctions vanish, while at

1

their bounded values and bounded first derivatives multiply a factor tending to zero. The right side is zero. Distinct eigenvalues therefore give the orthogonality of Sturm-Liouville eigenfunctions

\int_{1}^{R} x^{2} y_{i} (x) y_{j} (x) d x = 0 (λ_{i} \neq = λ_{j}) .

For complex eigenfunctions, replace

y_{i}

\overline{y_{i}}

, in accordance with the weighted inner product.

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