Q:

solution of a Cauchy problem (unique existence)

Prove unique existence of  the solution $u(t)$ of the following Cauchy problem:
\begin{cases} 
u'(t) = t(\frac{\pi}{3}-\arctan y(t)) \\
u(0) = 2
\end{cases}
$f(t,y)=t(\frac{\pi}{3}-\arctan y(t))\\$ 
$f \in C^0(R^2)$ so for the Peano  theorem the solution exists locally  $\\$
$\partial y f=-\frac{x}{1+y^2} $ , $\partial y \in C^0(\mathbb{R}^2)$, $f\in C^1(\mathbb{R}^2)$ 
so $f$ is locally lipschitz for $y$ uniform in $t.\\$
For Cauchy-Lipschitz theorem I have unique existence locally.
how can I prove global existence in $\mathbb{R}$? 

A:

Your $f$ is globally Lipschitz in $y$ on any fixed finite time interval $[-T,T]$ with $T>0$. Thus each step of Picard iteration will increase the size of your interval of definition of the solution by a fixed amount (something like $1/L$ where $L$ is your Lipschitz constant) provided that you remain within $[-T,T]$. Thus you can eventually fill out $[-T,T]$ with your solution regardless of what $T$ is. This gives global existence.

