Existence of unique solutions to a first order ODE on this interval

[thidmir]

TL;DR Summary

I am trying to find if there is a way to prove the existence and uniqueness of a solution
to a first order ODE on an interval including infinity.

I am trying to find a way to prove that a certain first order ode has a unique
solution on the interval (1,infinity). Usually the way to do this is to show that
if x' = f(t,x) (derivative with respect to t), then f(t,x) and the partial derivative with respect to f are continuous.
However, this would show that a solution exists only on an interval inside (1,infinity).
Is there any way to show that a solution exists on the entire interval?

 

[wrobel]

In general a solution is not obliged to be defined for all $t\ge 0$. For example $$\dot x=x^2,\quad x(0)=1$$
If in equation $\dot x=f(t,x)$ the function $f$ is such that $$|f(t,x)|\le c_1+c_2|x|,\quad (t,x)\in\mathbb{R}_+\times\mathbb{R}^m$$ then all the solutions to such a system are defined in $[0,\infty)$. There are a lot of other different sufficient conditions for that

 

[wrobel]

The following theorem is also useful.
Assume that $$f(t,x)\in C^1((t_1,t_2)\times D,\mathbb{R}^m)$$ where $D\subset\mathbb{R}^m$ is an open domain.

Assume also that $$|f(t,x)|\le c$$ for all $(t,x)\in (t_1,t_2)\times D$.

Theorem. Let $x(t)$ be a solution to the following IVP
$$\dot x=f(t,x),\quad x(t_0)=x_0\in D,\quad t_0\in(t_1,t_2).$$ Assume that $x(t)$ is defined in $[t_0,t^*),\quad t^*<t_2$ and can not be extended longer than $t^*$. Then the following limit exists
$$\lim_{t\to t^*-}x(t)=x^*$$ and $$x^*\notin D.$$