# Equilibrium solutions

### Equilibrium solutions

#### Lessons

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
• Introduction
What are equilibrium solutions?

• 1.
Finding Equilibrium Solutions
Find all the following equilibrium solutions for the following autonomous equation:

$\frac{dy}{dt}=y^2-3y-4$

Classify each equilibrium solution as either stable or unstable

• 2.
Finding Equilibrium Solutions
Find all the following equilibrium solutions for the following autonomous equation:

$\frac{dy}{dt}=(\frac{1}{5})(y^2-1) (y+3)^2$

Classify each equilibrium solution as either stable or unstable