**Equilibrium Solutions** are solutions to differential equations where the derivative equals zero along that solution. I.e. the slope is a horizontal line at that solution.

Note the Logistic Equation:

$\frac{dP}{dt}=KP(1-\frac{P}{M})$
With

$K$ and

$M$ being constants. This is a function of

$P$.

$\frac{dP}{dt}=f(P)$
This is an example of an

__Autonomous Differential Equation__.

An

**Autonomous Differential Equation** is a differential equation that is of the form:

$\frac{dy}{dt}=f(y)$
If we can find a solution such that

$f(y)=0$ for some

$y$, then this will be an

__Equilibrium__
__Solution.__
A

**Stable Equilibrium Solution** is an equilibrium solution that all solutions “near” to this equilibrium solution

__converge__ on it.

An

**Unstable Equilibrium Solution** is an equilibrium solution that all solutions “near” to this equilibrium solution

__diverge__ from it