= Solution
If $f$ has no explicit $x$ dependence, the <chain rule> gives
$$
\frac{df}{dx}=f_y y'+f_{y'}y''.
$$
Consequently
$$
\frac d{dx}(f-y'f_{y'})=y'\left(f_y-\frac d{dx}f_{y'}\right)=0
$$
on an <Euler-Lagrange equation> solution. Thus the <Beltrami identity> supplies the <first integral>
$$
\boxed{h=f-y'\frac{\partial f}{\partial y'}=\text{constant}.}
$$
The opposite sign is an equally valid convention. The conserved value can be zero for particular integrands or paths; absence of explicit $x$ alone does not guarantee a nonzero numerical value. The useful statement is that this expression is constant along an extremal.
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