Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 13D / a

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- Solution a

Solution

y + ε η

with

η = η^{'} = 0

at both endpoints, integrating the

η^{'}

term once and the

η^{''}

term twice gives

δ I = \int_{a}^{b} (f_{y} - \frac{d}{d x} f_{y^{'}} + \frac{d ^{2}}{d x ^{2}} f_{y^{''}}) η, d x .

The fundamental lemma yields the higher-order Euler-Lagrange equation

f_{y} - \frac{d}{d x} f_{y^{'}} + \frac{d ^{2}}{d x ^{2}} f_{y^{''}} = 0 .

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