Consider the ordinary differential equation $$\frac{dy}{dx}=f(y(x)), y(0)=a\in\mathbb{R}$$ where $f:\mathbb{R}\rightarrow \mathbb{R}$ be a real function. My question is that can i say that the maximal interval for solution of the above problem is $\mathbb{R}$ if $f$ is bounded and continuously differentiable? I tried with several example this and according to me its seems to be true. Please suggest me appropriate proof. Thanks.
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If $|f(y)|<M$ globally, then for any solution $|y(x)-y(0)|\le M|x|$. Thus a blow-up in finite time is impossible, the solution has to exist on all of $\Bbb R$.
Lutz Lehmann
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you shows its globally Lipschtz? – neelkanth Mar 25 '19 at 16:56
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Yes, that is another consequence of the boundedness of $f$, as that then also bounds the derivative of $y$. – Lutz Lehmann Mar 25 '19 at 17:57