Let us define this Cauchy Problem:
$$\frac{d x(t)}{dt}=(1+t^2)(\exp(x(t)-1)) \cos(x^2(t)), \quad x(0)=1, (t,x)\in \mathbb{R^2}$$
I have tried to show that ODE have a unique maximal Solution $\phi$ over $I=]T_{1},T_{2}[$ of class $C^1$ using the Cauchy-Lipchitz theorem but I didn't succeed.
And what is the limit of this solution for $t \to \pm\infty$?