Past exam of the mathematical methods I course of the University of Oxford / 2022 / 11 / b

On the component $x>0$ containing the boundary point, divide by $2x$ and use integrating factor $x^{-1/2}$:Integration gives $y=x^3/5+C\sqrt{x}$, and $y(1)=1$ fixes $C=4/5$. ThereforeAlthough the function has a continuous limit at zero, its derivative diverges there; the solution with this boundary datum does not extend as a differentiable solution through the singular point of the equation.