 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-1/9/a/solution

If the integrand has no explicit

x

dependence, differentiate

f - y^{'} f_{y^{'}}

along a stationary path:

\frac{d}{d x} (f - y^{'} f_{y^{'}}) = f_{x} + y^{'} (f_{y} - \frac{d}{d x} f_{y^{'}}) = 0.

The last equality uses

f_{x} = 0

f - y^{'} f_{y^{'}} = C

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