= Logarithmic change of variable in a Sturm-Liouville problem
{title2=$t=\log x,\quad x^2y''+xy'=Y''(t)$}
For a <Cauchy-Euler equation>, logarithmic coordinates turn scale changes in $x>0$ into translations in $t$. With $Y(t)=y(e^t)$, the <chain rule> gives $xy'=Y'$ and $x^2y''=Y''-Y'$. Consequently the <Sturm-Liouville operator> $-x^2y''-xy'-cy$ becomes $-Y''-cY$. Its natural <weighted inner product> has measure $dx/x=dt$, so this change of variable also preserves the norm when the transformed problem uses ordinary integration in $t$. On $1<x<e^\ell$ with <Dirichlet boundary conditions>, the orthonormal <eigenfunctions> are $\sqrt{2/\ell}\sin(n\pi\log x/\ell)$ and the <eigenvalues> are $(n\pi/\ell)^2-c$, for $n\geq1$.
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