= Hyperbolic reduction of a regular-singular differential equation
{title2=$y=xu\ \Longrightarrow\ u''-u=0$}
The equation $x^2y''-2xy'+(2-x^2)y=0$ has a <regular singular point> at zero. Factoring $y=xu$ cancels the Euler derivative terms and leaves $u''-u=0$ away from zero. Its analytic solutions $x\cosh x$ and $x\sinh x$ extend across zero. Their coefficient recurrence $(n-1)(n-2)c_n=c_{n-2}$ has independent free coefficients $c_1,c_2$, giving two entire solutions without a logarithmic branch despite integer-separated indicial roots.
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