Theorem (Picard-Linedlöf)

Let $U\subset \mathbb{R}^{n}$ be an open set, let $f:U\to \mathbb{R}^{n}$ be Lipschitz in $U$ with constant $k\ge 0$. Let $x_{0}\in U$ and

1. $r>0$ small enough s.t. $$\overline{B(x_{0},r)}=\{ x\in\mathbb{R}^{n}:\lVert x-x_{0} \rVert\le r \}\subset U$$
2. $$M=\sup_{x\in\overline{B}(x_{0},r)} \lVert f(x) \rVert$$
3. $a>0$ s.t. $a\le \frac{r}{M}$ and $a< \frac{1}{k}$

Then $\forall t_{0}\in\mathbb{R}, \exists!f\in C^{1}([t_{0}-a,t_{0}+a];\overline{B(x_{0},r)})$ s.t. $$\begin{cases} \dot{x}(t)&=f(x(t))\\ x(t_{0})&=x_{0} \end{cases},\quad t\in[t_{0}-a,t_{0}+a]$$ i.e. there exists a unique solution to our initial value problem.