$$y'(t)=e^{y(t)}+\cos y(t)+\sqrt{t\,}\,y^2(t)$$ $$y(0)=1$$.
I know it has at least one solution using Peano's theorem. I know $y(t)$ is increasing and positive, and I also found the maximum interval on which the solution can be defined is $U=\Bigl[0,\bigl(\frac{3}{2}\bigr)^{2/3}\Bigr]$ and then it blows up at the $\bigl(\frac{3}{2}\bigr)^{2/3}$. Now for any given rectangle $$R=[0,a]\times[1-b,1+b]$$ where $[0,a]$ is a subset of $U$ i have that $δ(y,t)=e^{y(t)}+\cos(y(t)+\sqrt{t\,}\,y^2(t)$ is Lipschitz since $δ_y(y,t)=e^{y(t)}-\sin y(t)+2\sqrt{t\,}\,y(t)$ has a max value on that rectangle. So in subset intervals it has a unique solution, but what can I say for the whole $U$? Since it blows up it cannot be Lipschitz in the whole $U$ but that DOESN'T mean its solution is not unique; it just means I can't use Picard's theorem.