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:

dPdt=KP(1PM)\frac{dP}{dt}=KP(1-\frac{P}{M})

With KK and MM being constants. This is a function of PP.

dPdt=f(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:

dydt=f(y)\frac{dy}{dt}=f(y)

If we can find a solution such that f(y)=0f(y)=0 for some yy, 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
  • 1.
    What are equilibrium solutions?

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

    dydt=y23y4\frac{dy}{dt}=y^2-3y-4

    Classify each equilibrium solution as either stable or unstable

  • 3.
    Finding Equilibrium Solutions
    Find all the following equilibrium solutions for the following autonomous equation:

    dydt=(15)(y21)(y+3)2\frac{dy}{dt}=(\frac{1}{5})(y^2-1) (y+3)^2

    Classify each equilibrium solution as either stable or unstable