= Solution
The useful coordinate for this scale-dependent differential operator is $t=\ln x$. With $Y(t)=y(e^t)$, the <chain rule> gives
$$
xy'=Y_t,\qquad x^2y''=Y_{tt}-Y_t,\qquad Ly=-Y_{tt}-Y.
$$
The interval becomes $0\leq t\leq1$, and the endpoint values become $Y(0)=Y(1)=0$. This <logarithmic change of variable in a Sturm-Liouville problem> reduces the calculation to a constant-coefficient Dirichlet problem. Also $dx/x=dt$, so the transformation preserves the appropriate <weighted inner product>.
Back to article page
