Assume the IVP
\begin{align} \begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix}' &= \begin{bmatrix} x_{1}^2 \\ x_{2}+x_1^{-1} \\ \end{bmatrix}, \quad\begin{bmatrix} x_{1}(0) \\ x_{2}(0) \\ \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ \end{bmatrix} \end{align}
Writing it as $$ \vec{x}'=\vec{F}(x)$$ we can see that, since all the partial derivatives of $F$ are continuous within a compact interval $(-\delta,\delta)$ near $t=0$ and thus bounded, $F$ satisfies Lipschitz condition locally. So Picard says that there exists a unique solution within this neighborhood.
Also, another theorem states that if $F$ is continuous in some $D$ and $x(t)$ is the solution of the equation for some interval $J$, then $x(t)$ can be extended to a maximum interval $J^* \supset J$.
How can I find the maximum existence interval of this problem's solution?