Finding interval of definition of ODE solutions.

Question

I would like to know how the interval of definition for an initial value problem be determined without explicitly solving it. For example, the solution to $$y' = e^y + 1, \; y(0) = 0$$ is $$ y(t) = \log \left( \frac{e^t}{2 - e^t} \right)$$ hence the interval of definition $$t < \log(2)$$ And also in the case of higher order ODEs, for example in, $$y'''+\frac{\cos{3t}}{4-t}y'+\frac{\sin{2t}}{5-t}y = \frac{e^{-2t}}{1+t}, \; y(2) = y'(2) = y''(2) = 0$$ is there way you can determine the interval of definition?