I've been trying to grasp some theorems on the extension of solutions and I still have some questions.
The theorems say thing about the solution escaping compact sets, being unbounded, etc. but I'm having trouble applying them to concrete equations.
Is there any way to estimate the size of the interval of a solution?
E.g., given the equation:
$x' = 1 + x^2$
how can we know, without explicitly or numerically solving it, that the interval where its solutions are defined has finite longitude? This is rather easy to see if you solve the equation as you get
$x(t) = tan(c + t)$
which is only defined in intervals of finite longitude.
But how would one reach the same conclusion just studying $f(x) = 1 + x^2$ ? Is this even possible?
On a related note, how does one go about proving that a solution can be extended to infinity?
Thank you in advance.