I am looking at an ODE of the form \begin{align} \dot{x} = f(x). \end{align} Do poles of $f$ give me boundaries for the maximal solution $\lambda(t)$ of an ODE of this form?
For example:
\begin{align} \dot{x} = \frac{1}{\sin(x)},\qquad x(0) = \frac{\pi}{2}. \end{align} Obviously $x = 0$ and $x = \pi$ are poles of $\frac{1}{\sin{x}}$. Can I conclude that the maximal solution $\lambda$ of this initial value problem fulfills $0 < \lambda(t) < \pi$ for all $t$ of the maximal interval of existence? In this particular case: Does it allow me to conclude that the maximal solution exists for all $t\in \mathbb{R}$?
Yes, the solution must be continuous, and it cannot take a value which makes $\dot x$ undefined, so it can't go outside the interval $(0,\pi)$.
But no, you can't conclude that the maximal interval of existence is $\mathbb{R}$, since if you simply solve the ODE using separation of variables, you get $$ x(t) = \arccos(-t) ,\quad -1 < t < 1 . $$