Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 17 / b / Solution

Away from $x=-1$ the separable differential equation givesOn the interval containing $x=1$, one has $x>-1$, so the positive initial value selects $y=K(1+x^3{)}^{1/3}$. Substitution of $y(1)=2$ gives $K=2^{2/3}$. ThusThe zero solution omitted by division by $y$ cannot satisfy the initial value. Although the selected solution tends to zero as $x\downarrow -1$, its derivative diverges there; it does not extend through that point as a classical differentiable solution.