= Power-coordinate reduction of a singular Sturm-Liouville quotient
{c}
{title2=$z=x^{3/2},\quad I(w)=\frac92\frac{\int_0^1(W')^2dz}{\int_0^1W^2dz}$}
The singular equation $2xw''-w'+\lambda x^2w=0$ becomes $W''+(2\lambda/9)W=0$ under $z=x^{3/2}$, $w(x)=W(z)$. Its zero-endpoint finite-<energy> quotient transforms as $2\int_0^1x^{-1/2}(w')^2dx/\int_0^1x^{1/2}w^2dx=(9/2)\int_0^1(W')^2dz/\int_0^1W^2dz$. The <Poincaré inequality> therefore proves the sharp minimum $9\pi^2/2$, attained by $w=\sin(\pi x^{3/2})$. This coordinate change turns an apparently singular weighted variational problem into the usual interval <energy> problem. The admissible eigenfunction is $O(x^{3/2})$, making the integration-by-parts <boundary term> $wx^{-1/2}w'$ vanish and both <energy> <integrals> converge.
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