Hyperbolic reduction of a regular-singular differential equation ($y=xu⟹u^{\prime \prime }-u=0$)

The equation $x^2{y}^{\prime \prime }-2x{y}^{\prime }+(2-x^2)y=0$ has a regular singular point at zero. Factoring $y=xu$ cancels the Euler derivative terms and leaves $u^{\prime \prime }-u=0$ 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.