= Jacobi identity for the second variation
{c}
{title2=$P\varphi'^2+Q\varphi^2=P(\varphi'-w'\varphi/w)^2+(Pw'\varphi^2/w)'$}
If $w$ is a nonvanishing solution of $-(Pw')'+Qw=0$, expanding the right-hand side gives the displayed identity. Integrating expresses a scalar <second variation> as a square plus a boundary term. A simple zero of $w$ at a fixed endpoint causes no difficulty for <Lipschitz> endpoint-vanishing variations, since $\varphi=O(x-a)$ makes the boundary term tend to zero. This identity is the scalar form of a <ground-state transform for a weighted Dirichlet energy>.
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