= Solution
Let $y_\varepsilon=y+\varepsilon\eta$, where the <variation> $\eta$ satisfies $\eta(a)=\eta(b)=0$. Differentiating the <functional> at $\varepsilon=0$ gives its <first variation>
$$
\delta I
=\int_a^b\left(L_y\eta+L_{y'}\eta'\right)\,dx.
$$
Using <integration by parts> and the fixed endpoint conditions,
$$
\delta I
=\int_a^b\left(
L_y-\frac d{dx}L_{y'}
\right)\eta\,dx.
$$
If this vanishes for every admissible $\eta$, the <fundamental lemma of the calculus of variations> gives the <Euler-Lagrange equation>
$$
\boxed{\frac d{dx}\frac{\partial L}{\partial y'}
-\frac{\partial L}{\partial y}=0}.
$$
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