TITLE: Blow-up of ODE solution
QUESTION [6 upvotes]: I am a newcomer to ODEs. The relevant theorem that I can think of is about the maximum open interval of existence of the solution. But I have not learned to find the interval on which the solution exists.
Let $f : \mathbb{R}^{n}\to \mathbb{R}^{n}$ be $C^{1}$ and bounded
on $\mathbb{R}^{n}$. Is it possible to have a solution of $\dot{x}=f(x)$
that blows up in finite time?

REPLY [8 votes]: No. Since $x(t) = x_0 + \int_0^t f(x(\tau)) \, d \tau$, and $f$ is bounded by, say, $B$, you have
$$ \|x(t)\| \leq \|x_0\| + \int_0^t \|f(x(\tau))\| d \tau \leq \|x_0\| + t B.$$
Hence $x(t)$ is bounded when the time is bounded.