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 $f-y^{\prime }{f}_{y^{\prime }}$ along a stationary path:The last equality uses $f_x=0$ and the Euler-Lagrange equation. This proves the Beltrami identity, the first integral $\boxed{f-y^{\prime }{f}_{y^{\prime }}=C}$.