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

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- Solution b

Solution

Differentiate

Q = f - (f_{y^{'}} - \frac{d}{d x} f_{y^{''}}) y^{'} - f_{y^{''}} y^{''} .

The product rule cancels all terms except

Q^{'} = y^{'} (f_{y} - \frac{d}{d x} f_{y^{'}} + \frac{d ^{2}}{d x ^{2}} f_{y^{''}}) + f_{x} .

On an extremal with no explicit

x

dependence both terms vanish, proving the stated conserved quantity.

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