Quadratic-variable reduction of a singular oscillator ($x{y}^{\prime \prime }-y^{\prime }+4x^{3}y=0,u=x^2$)

The chain rule with $y(x)=Y(x^2)$ gives $x{y}^{\prime \prime }-y^{\prime }=4x^3{Y}^{\prime \prime }$ and transforms the equation, away from zero, into $Y^{\prime \prime }+Y=0$. 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 $y^{\prime \prime }(0)$ to select the two solutions; every smooth solution has $y^{\prime }(0)=0$.