= Vanishing prefactor at one time does not ensure a uniform Lipschitz bound
{title2=$f(x,y)=x|y|^{2/3}$}
A coefficient that vanishes at the initial time can hide failure of the uniform <Lipschitz condition> required for an <initial value problem>. For $f(x,y)=x|y|^{2/3}$, the map $y\mapsto f(0,y)$ is identically zero. But every rectangle of positive widths around $(0,0)$ contains some $x\neq0$, and
$$
\frac{|f(x,y)-f(x,0)|}{|y|}=|x||y|^{-1/3}\longrightarrow\infty.
$$
No Lipschitz constant works uniformly on the rectangle. Actual uniqueness also fails: $y=0$ and $y=x^6/216$ are distinct differentiable solutions with initial value zero. Substitution gives $y'=x^5/36=x|y|^{2/3}$ for the second solution. A condition at the single initial time cannot replace control on a neighbourhood.
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