Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2007 / 7 / iii / Solution

Translation in $x$ leaves this autonomous differential equation unchanged. For every $c\ge 1$, the same formulasatisfies $u_c(1)=0$, and the zero function does too. Thus solutions exist and are not unique, exactly as for the initial point $(0,0)$. The relevant failure is the Lipschitz condition at $y=0$; moving the initial value of $x$ does not remove it.