Lipschtiz continuity for inital value problem $u'(t) = u(t)^{2/3}, u(0) = 0$

Question

I was asked to show that the solution for this initial value problem is not unique, first just solving the equation and then using the uniqueness theorem. For the first case the have the trivial solution $u_{0}(t) = 0$ and then separating variables and using the fact that $u_{1}(t) {\not=} 0$ we get $u_{1}(t) = (c + t)^3$.

Now the problem is when I have to show this via the uniqueness theorem, I need to show that the function is not Lipschtiz continuous in some interval including $t = 0$ and I'm not sure how to do that.

Thanks in advance

Answer 1

Hint: If $f'(t)$ is not bounded in any neigborhood of $t=0$, then $f$ is not Lipschitz.