Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 1 / a

With the change of variables $z=y^2$, multiplication of the differential equation by $2y$ gives $z^{\prime }+2xz=2x$. The integrating factor is $e^{x^2}$, soThe initial condition gives $C=3$. Since $y$ starts negative and $z\ge 1$, continuity selects the negative square root throughout:This never reaches the singular value $y=0$.