### The stability of equilibria of a differential equation

*The stability of equilibria of a differential equation.*

*The stability of equilibria of a differential equation, analytic approach.*

#### Stability theorem

Let $\diff{x}{t} = f(x)$ be an autonomous differential equation. Suppose $x(t)=x^*$ is an equilibrium, i.e., $f(x^*)=0$. Then

- if $f'(x^*)<0$, the equilibrium $x(t)=x^*$ is stable, and
- if $f'(x^*)>0$, the equilibrium $x(t)=x^*$ is unstable.

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