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

Question

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.

Answer 1

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

• Existence of solution $y(x)$ with $x \in [0,\frac12]$
• Riccati D.E., vertical asymptotes
• Prove that the IVP $\begin{cases}\dot{x}=x^3+e^{-t^2}\\x(0)=1\end{cases}$ has an unique solution defined on $I=(-1/9,1/9)$