In my course notes, we are working on the stability of solutions, and in one example we start out with:
Consider the IVP on $(-1,\infty)$:
x' = \frac{-x}{1 + t} with $x(t_{0}) = x_{0}$.
Integrating, we get $x(t) = x(t_{0})\frac{1 + t_{0}}{1 + t}$.
I can't produce this integration but the purpose of the example is to show that $x(t)$ is uniformly stable, and asymptotically stable, but not uniformly asymptotically stable.
But I can't verify the initial part and don't want to just skip over it.
Can someone help me with the details here?
Update: the solution has been pointed out to me and is in the answer below by Bill Cook (Thanks!).