= Solution
Vary $y$ by $y+\varepsilon\eta$, where the clamped endpoint conditions require $\eta=\eta'=0$ at both endpoints. The <first variation> of the quadratic <elastic energy> is
$$
\delta E_y=\int_0^L\bigl(Ay''\eta''-Fy'\eta'\bigr)\,dx
=\int_0^L\bigl((Ay'')''+Fy''\bigr)\eta\,dx.
$$
Both integrations by parts have zero boundary terms. Since the variation is arbitrary in the interior, the <fundamental lemma of the calculus of variations> gives the <Euler-Lagrange equation>. Varying $z$ independently gives the second equation:
$$
\boxed{(Ay'')''+Fy''=0,\qquad(Bz'')''+Fz''=0.}
$$
The clamped endpoint values accompany these equations. \b[If $A$ or $B$ depends on $x$, it must remain inside the derivatives.]
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