Prove that maximal solutions are defined in $mathbb{R}$
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Let $f:mathbb{R}timesmathbb{R}^nto mathbb{R}^n$ be a continuously differentiable function. Suppose that exists $v:mathbb{R}to[0,infty)$ continuous such that $||f(t,x)||leq v(t)||x||; forall (t,x)in mathbb{R}timesmathbb{R}^n$ . Prove that all maximal solutions for $dot{x}=f(t,x)$ are defined in $mathbb{R}$ ; what happens if $f$ is bounded? I believe that the Picard–Lindelöf theorem is the way to go, but I don't know how to prove that $f$ satisfies the Lipschitz condition. In order to use it, if I understand correctly, I should prove that $v$ is bounded for every $t$ in a neighborhood of $(t_0,x_0)$ being the last a solution for the differential equation. If $dot{x}=f(t,x)$ then $x(t)=displaystyleint f(t,x)dt$ . Considering the hypothesis, $$displaystyle{||x||leqint||f(t,x)||dtleqint v(t) ||x||dt