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 $\forall t\in \mathbb R$.


Answers can be viewed only if
  1. The questioner was satisfied and accepted the answer, or
  2. The answer was disputed, but the judge evaluated it as 100% correct.
View the answer
  • There is a typo in the third paragraph, I meant there exists delta_2>0 and not delta_1.

The answer is accepted.