Solution
= Solution
For variations $y+\varepsilon\eta$ with $\eta=\eta'=0$ at both endpoints, integrating the $\eta'$ term once and the $\eta''$ term twice gives
$$
\delta I=\int_a^b
\left(f_y-\frac d{dx}f_{y'}+\frac{d^2}{dx^2}f_{y''}\right)\eta,dx.
$$
The fundamental lemma yields the higher-order <Euler-Lagrange equation>
$$
\boxed{f_y-\frac d{dx}f_{y'}+\frac{d^2}{dx^2}f_{y''}=0}.
$$