I am trying to solve the following Cauchy problem:
$$ y' = \sin(y), \quad y(0) = \frac{\pi}{3} $$
After some calculation, I get to
$$ \int_{\frac{\pi}{3}}^y \frac{ds}{\sin{s}} = x $$
Actually, I just want to find the maximum interval where the solution is defined, so I study the integral function: $$ F(y) = \int_{\frac{\pi}{3}}^y \frac{ds}{\sin{s}} $$
Now I know I have to calculate some limits to find the range, but I am not sure how. Can someone give me some hints?
First of all, notice that $y = 0$ and $y = \pi$ are trivially solutions to the differential equation. Since the graphs of two different solutions cannot cross eachother, this implies that the solution going through $(0,\pi/3)$ must satisfy $0 < y(x) < \pi$ for all $x$ where it is defined.
Furthermore, $y'(x) > 0 $ for all points in this region, so the solution is a monotonically increasing function of $x$. You can decuce all properties of the solution you need from these two observations.