I would appreciate if someone could please give me a hint on how to do this problem. Or where to see some examples. Unfortunately, the sources that I have do not seem to actually explain it and show some examples.
Given the IVP $x'=\cos^5(x) +1$, $x(0)=x_0$, show that this IVP has a unique solution for all $t\in \mathbb{R}$.
The right-hand side is $C^1$ and so it is continuous and locally Lipschitz. Thus, any initial value problem has a unique solution, by the Picard-Lindelöf theorem (the original result by Picard had an additional assumption, and thus why one does not use universally "Picard theorem").
Moreover, in this case the derivative is bounded, and so the right-hand side is in fact Lipschitz (and not only locally Lipshitz). A simple application of Gronwall's lemma shows that all solutions are global (that is, are defined on the whole line).