= Quadratic-variable reduction of a singular oscillator
{title2=$xy''-y'+4x^3y=0,\qquad u=x^2$}
The <chain rule> with $y(x)=Y(x^2)$ gives $xy''-y'=4x^3Y''$ and transforms the equation, away from zero, into $Y''+Y=0$. Thus its solution space is spanned by $\cos(x^2)$ and $\sin(x^2)$. Both extend smoothly through the <regular singular point> at zero. Their even <power series> allow the independent data $y(0)$ and $y''(0)$ to select the two solutions; every smooth solution has $y'(0)=0$.
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