An ODE is autonomous when the independent variable does not appear explicitly:

$$\frac{dy}{dt}=f(y).$$

An equilibrium is a value $y_*$ satisfying

$$f(y_*)=0.$$

If the system starts exactly at an equilibrium, it remains there.

Away from equilibria, the sign of $f(y)$ determines the direction of motion. Where $f(y)>0$, solutions increase; where $f(y)<0$, they decrease.

A phase line records these directions on the $y$-axis. An equilibrium is locally stable when nearby trajectories move toward it and unstable when they move away.

This qualitative viewpoint can determine long-term behavior without finding an explicit formula for $y(t)$.