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If you take a random messy Cauchy problem like $$y'=y^3+xy+e^x,\quad y(0)=1$$ Chances are it will explode (if it does) at a some $x$ values that are nothing special, like $-2.8172...$ and $1.2234...$. Recently I posted about an ODE that seemed hard to solve, at least at my level, because it involved some Bessel functions: $$y'=y^2+x^2,\quad y(0)=0.$$ This Cauchy problem however caught my attention because it seemed to explode at $\pm 2$, something I found odd given the relative complexity of the solution... but in the end the maximal interval turned out to be an unexpected $]\pm 2.003...[$.

This got me thinking: are there any Cauchy problems that are hard (or yet!) to solve but of which we know for certain the maximal interval?

I'm especially talking about maximal intervals that are not $\mathbb{R}$ (boring!) and in particular those who take a pretty form, i.e. with rational/cute numbers as extrema.

aleph2
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  • The maximal interval also depends on the initial value, dynamic blow-up in contrast to static blow-up. This behavior one can already find in the standard examples $y'=y^2+c$, $c=cont.$ – Lutz Lehmann Mar 19 '24 at 16:04
  • But those are easy to solve. My question is about whether there are hard-to-solve / unsolved CPs of which we nevertheless know the maximal interval. – aleph2 Mar 19 '24 at 16:08

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