Which properties/tools can I use to check whether the differential equation $y'(x) = x^5(e^{4-y^2}-1)$ has a global existence of solutions?
I know that for globally Lipschitz, the derivative of the function needs to be bounded but I don't know how can I bound such a function. Anyone who can help me out with this?
You have a differential equation $y'=f(x,y)$ where $f$ is $\mathcal{C}^1$, so Cauchy - Lipschitz theorem applies. Moreover, if you had a maximal non-global solution defined on an open interval $]\alpha,\beta[$ with $\beta$ finite, say, then a theorem (in french, « théorème des bouts ») ensures that every maximal non-global solution leaves every compact, namely (here) $y(x)$ would go to infinity when $x$ goes to $\beta$.
In your example, this cannot happen since because of the inequality $|y'| = |f(x,y)| \le e^4|x|^5$, which ensures that every solution is Lipschitz on bounded intervals.