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I am a newcomer to ODE. 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.

$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is $C^{1}$ and bounded on $\mathbb{R}^{n}.$ Is it possible to have a solution of $\dot{x}=f(x)$ that blows up in finite time?

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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.

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