# Math Insight

### The stability of equilibria of a differential equation

The stability of equilibria of a differential equation.

Imagine that, for the differential equation \begin{align*} \diff{x}{t} &= f(x)\\ x(0) &=b \end{align*} where $f(-2.7)=0$, you determine that the solution $x(t)$ approaches $-2.7$ as $t$ increases as long as $b < -2.5$, but that $x(t)$ blows up if the initial condition $b$ is much larger than $-2.5$. What can you conclude about the point $x=-2.7$?

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.