= Solution
The monotonic horizontal coordinate allows the path to be written as a graph. Its <arc length> element is $ds=\sqrt{1+y'^2}\,dx$, so the variational density is $f(y,y')=g(y)\sqrt{1+y'^2}$. It has no explicit horizontal-coordinate dependence. The <Beltrami identity> becomes
$$
f-y'f_{y'}=g\sqrt{1+y'^2}-\frac{g y'^2}{\sqrt{1+y'^2}}
=\frac{g}{\sqrt{1+y'^2}}.
$$
Consequently every smooth minimizing path satisfies
$$
\boxed{\frac d{dx}\left(\frac{g(y)}{\sqrt{1+y'^2}}\right)=0.}
$$
This is a first integral of the weighted arc-length problem; fixed endpoint values remove the endpoint variation terms.
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