= Solution
{c}
Let $P=F_{y'}$ and $Q=F_{y''}$. For a smooth <variation> $y+\varepsilon\eta$, the <first variation> is
$$
\delta J=\int_a^b(F_y\eta+P\eta'+Q\eta'')\,dx
=\int_a^b(F_y-P'+Q'')\eta\,dx+[(P-Q')\eta+Q\eta']_a^b.
$$
Here the last expression follows by two integrations by parts. Taking variations supported in the interior and applying the <fundamental lemma of the calculus of variations> gives the <Euler-Lagrange equation for a second-derivative functional>:
$$
\boxed{F_y-\frac{d}{dx}F_{y'}+\frac{d^2}{dx^2}F_{y''}=0.}
$$
If both endpoint values and slopes are free, their independent variations force the <natural boundary conditions for second-derivative functionals>
$$
\boxed{Q=0,\qquad P-Q'=0\quad\text{at }a,b.}
$$
If an endpoint value or slope is fixed instead, its corresponding variation vanishes and the associated natural condition is not required.
For $H=y''Q-y'(Q'-P)-F$, differentiate using the <chain rule>:
$$
H'=y'(P'-Q''-F_y)-F_x.
$$
The <Euler-Lagrange equation> cancels the first term. Consequently $H'=-F_x$, and if there is no explicit $x$ dependence,
$$
\boxed{H=y''F_{y''}+y'\bigl(F_{y'}-(F_{y''})'\bigr)-F\text{ is constant}.}
$$
This is a <Beltrami identity for a second-derivative functional>.
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