Solution
= Solution
For a variation $y+\varepsilon\eta$ with $\eta(a)=\eta(b)=0$, the first variation of the <functional> is
$$
\delta I=\int_a^b(F_y\eta+F_{y'}\eta')dx=\int_a^b\left(F_y-\frac{d}{dx}F_{y'}\right)\eta\,dx.
$$
The boundary term vanishes by the fixed endpoints. The <fundamental lemma of the calculus of variations> therefore gives the <Euler-Lagrange equation>
$$
\boxed{\frac{d}{dx}F_{y'}-F_y=0.}
$$