# How do I study the domain of a Cauchy's problem without solving it?

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.

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.