Let
$\displaystyle y'(t)=\pm y^2(t), \ y(0)=y_0,\ y(t)\text{ real}$
be given.
I want to determine in which of the two cases $(\pm)$ the solution is global in $(0,\infty)$. Since my book discusses the Cauchy-Kowalewskaja (CK) Theorem, I assume this is what I have to work with.
Firstly, the solutions are
$\displaystyle y(t)=\frac{1}{1/y_0 \mp t}$
I know form CK Thm that, if I were to assume that $y^2(t)$ is real analytic near the origin and the solution $y(t)$ is unique, then also $y(t)$ is real analytic near the origin.
Now obviously, if the denominator can be zero in a neighbourhood of $0$ then the neighbourhood has to be made smaller. But since I want a global solution in $(0,\infty)$, my boundary condition is the critical point.
If for example we have $y_0>0$ then for the solution with the plus case, there should be no problems, right? Not really sure what to do from here.