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I have some problems that require to study the domain of the Cauchy's problem solution but I don't really know how to do that. For example,

$\begin{cases} y'=(y-\sin x)^2+1+\cos x\\ y(0)=0 \end{cases}$

I did few theorems about Cauchy's problem but none of them says where the solution is defined.

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Set $u=y-\sin x$, the the ODE reduces to $$ u'=u^2+1. $$ This has an easy solution via separation of variables with the then obvious maximal domain.

For problems where such easy solutions are not possible, see for instance

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  • $\begingroup$ Ok, thank you.. I don't think my course is that advance but still interesting. $\endgroup$ – Archimedess Dec 16 '18 at 11:35

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