= Euler-Cauchy classification from two regular singular points
{c}
{title2=$z^2y''+Azy'+By=0$}
Normalize the second-order equation as $y''+P(z)y'+Q(z)y=0$. If every point other than zero and infinity is ordinary and both exceptions are at worst <regular singular points>, then $zP$ and $z^2Q$ extend holomorphically across zero and stay bounded at infinity. <Liouville's theorem> makes them constants, yielding the displayed <Euler-Cauchy equation>. If both exceptions must be genuinely singular, exclude $(A,B)=(0,0)$ and $(2,0)$, which make zero and infinity respectively ordinary.
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