= Solution
Apply a constant <Lagrange multiplier> to the <isoperimetric constraint>, using the augmented functional $J-\lambda(I-1)$. Its integrand is
$$
F(x,y,y')=p(x)y'^2-q(x)y^2-\lambda w(x)y^2.
$$
The <Euler-Lagrange equation> gives $-2qy-2\lambda wy-2(py')'=0$, hence
$$
\boxed{(py')'+qy+\lambda wy=0.}
$$
The multiplier rule applies because the constraint derivative in direction $y$ is $DI_y[y]=2I[y]=2$, so the constraint derivative is not zero. The prescribed endpoint values remove boundary variations.
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