 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-1/13a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The Lagrange identity for this Sturm-Liouville theory operator is

y_{0} L (y; λ_{0}) - y L (y_{0}; λ_{0}) = \frac{d}{d x} [p (y_{0} y^{'} - y y_{0}^{'})] .

Integrating from zero to one, the boundary term vanishes because both

y

and

y_{0}

vanish at both endpoints. Since

L (y_{0}; λ_{0}) = 0

and

L (y; λ_{0}) = f

\int_{0}^{1} y_{0} f d x = 0.

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