Consider this first-order ODE $$ \dot{y}=f(x,y)=-y^{3}e^{2x}+4 $$
with $y(0)=y_{0}$.
I was able to verify existence of a solution. How can we prove uniqueness of solution for $x\geq0$? we know that $f$ is continuous in $x$, but I don't think that $f$ is global or local Lipschitz as I am unable to bound the $\frac{\partial f}{\partial y}$. Any suggestions?