Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2011 / Differential Equations / 7 / ii / Solution

No rectangle of positive width around $(0,0)$ has a uniform Lipschitz constant in $y$. For any nonzero $x$ in such a rectangle,The vanishing factor $x$ at the single initial point does not control neighbouring $x$-values. The failure of the hypothesis is accompanied by actual nonuniqueness: bothsatisfy the equation and have value zero at $x=0$. Direct differentiation gives $y_2^{\prime }=x^5/36=x(y_2{)}^{2/3}$, including the zero point by continuity.