Consider the scalar equation $$\dot{x}(t)=a(t)x(t)$$ which is stable and reducible.
I'm trying to show that $\dot{x}(t)=a(t)x(t)$ is uniformly stable when $a(t)\in C(t_0,\infty)$.
The precise definition for uniformaly stable. $$\forall \epsilon >0,\exists \delta = \delta (\epsilon)>0, \text{s.t}: \|x(t_0)\|<\delta \implies \|x(t)\|< \epsilon, \forall t \ge t_0 \ge 0$$
I want to use it to solve my problem. Any help will be appreciated! Thanks!
By separation of variables, for the solution $$x'/x = a \longrightarrow x(t)\ \to\ \text {x0} \ e^{\int_{t_0}^t a(\tau) d\tau} $$ remains to show $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{s.t} \ \ ||x_0|| < \delta \ \implies \ e^{\int_{t_0}^t a(\tau) d\tau} < e^{\max_{t_0,t}(a)(t-t_0)} < \varepsilon $$