Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 9 / iii

Here $F$ has no explicit $x$ dependence, so use the Beltrami identity. With $p=y^{\prime }$, one has $F_{y^{\prime }}=yp/\sqrt{1+p^2}$ and thereforeThis first integral reduces the second-order Euler-Lagrange equation to a first-order relation.