Show this initial value problem has a unique solution for initial value forall t

Given the initial value problem $u'=\cos(u)\sqrt{1+u^2}+e^{-u^2}$ and $u(0)=u_0$, show that it has a unique solution for each $u_0$ in $\mathbb R$ which exist $\forall t\in \mathbb R$.

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  • There is a typo in the third paragraph, I meant there exists delta_2>0 and not delta_1.

The answer is accepted.