= Solution
Take a smooth <variation> $y_\varepsilon=y+\varepsilon\eta$. Differentiating the <functional> at zero gives the <first variation>
$$
\delta I=\int_a^b(F_y\eta+F_{y'}\eta'+F_{y''}\eta'')\,dx.
$$
Integrate the second term once and the third term twice by <integration by parts>:
$$
\delta I=\int_a^b\left(F_y-\frac d{dx}F_{y'}+\frac{d^2}{dx^2}F_{y''}\right)\eta\,dx
+\left[\left(F_{y'}-\frac d{dx}F_{y''}\right)\eta+F_{y''}\eta'\right]_a^b.
$$
Choose variations supported away from the endpoints. Stationarity and the <fundamental lemma of the calculus of variations> force
$$
\boxed{F_y-\frac d{dx}F_{y'}+\frac{d^2}{dx^2}F_{y''}=0.}
$$
This is the <Euler-Lagrange equation for a second-derivative functional>. With the interior equation imposed, the remaining boundary expression must vanish for every admissible <variation>.
At each endpoint, a freely varying value $y$ requires $\boxed{F_{y'}-(F_{y''})'=0}$, and a freely varying slope $y'$ requires $\boxed{F_{y''}=0}$. A fixed value or slope instead makes its corresponding variation zero and supplies no natural condition for that variable. These are the <natural boundary conditions for second-derivative functionals>, with the value and slope variations treated independently.
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