= Solution
Multiply the <Sturm-Liouville equation> by $y$ and integrate. The endpoint values and <integration by parts> give
$$
0=\int_{x_0}^{x_1}\bigl[y(py')'+qy^2+\lambda wy^2\bigr]dx
=-\int_{x_0}^{x_1}(py'^2-qy^2)dx+\lambda I[y].
$$
Since $I[y]=1$,
$$
\boxed{J[y]=\lambda.}
$$
Thus the multiplier equals the stationary value of $J$ at the extremal, as in <stationary values of a quadratic functional with a quadratic constraint>. It is also the corresponding <Rayleigh quotient>.
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