For a differential equation like $\dot{x}=ax$ where $x$ is a function on an independent variable $t$, and $\dot{x}=\frac{dx}{dt}$, and $a$ is a constant, we define the time-scale $\frac{1}{a}$, which if $a$ be a negative real number we can look at this concept as the half-life time of $x$ over $\ln 2$. And you know half-life time means the time you need to loose half of your initial amount of $x$. As you can see it at the following calculation.

Let $a=-k$ where $k\in\mathbb{R}^{>0}$. From our differential equation we have $x=x\_0e^{-kt}$ so if I show the half-life time by $\tau\_{\frac{1}{2}}$ then
$$\frac{x\_0}{2}=x\_0e^{-k\tau\_{\frac{1}{2}}}\Longrightarrow \ln 2=k\tau\_{\frac{1}{2}}\Longrightarrow \tau\_{\frac{1}{2}}=\frac{\ln 2}{k}$$
With this concept **time-scale** we do many things, one is choosing the step size for numerical simulation of our differential equation using Euler method for example. Or if we have a system of differential equations in the form above, we say which one has a faster effect on our populations by comparing time-scales of each present differential equation.

But what should I define time-scale for a general form of a differential equation which at least do those two works for us that I mentioned?