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Let $g:[0, T]\times\mathbb{R}^n \to \mathbb{R}$, $a\in\mathbb{R}^n$. Consider the Cauchy problem $$\begin{cases}x'(t)=g(t,x) \quad a.e \quad s\in [0, T]\\ x(0)=a\end{cases}$$ where $g(t,x)$ is continuous in $t$, locally Lipschitz in $x$ and has linear growth in $x$. The problem has a unique solution $x(.)$ on $[0, T]$ with locally Lipschitz dependence on initial data.

But I can't find any uniqueness theorem for ODE to cite. Would you please give me a reference?

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You want the Picard-Lindelöf theorem. It guarantees both the local existence and the uniqueness of the solution. You can also use this to prove global uniqueness for your system.

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