Eigenvalues for Equilibrium Points of First Order Nonlinear DE

[jsi]

Homework Statement

dy/dx = y^3-3y^2+2y
it's asking for equilibrium points and for the eigenvalues and stability at each point.

Homework Equations

The Attempt at a Solution

I found the equilibrium points by setting dy/dx = 0 as we were taught to do in class and got y = 0, 1, 2. Then I took the derivative of the equation and evaluated it at each point to determine stability and found derivative at 0 = 2, at 1 = -1, and at 2 = 2 so the first and third are unstable and the second is stable. I'm confused on how to find eigenvalues for just one equation because I'm used to doing it for 2 equations. Have I already done it by evaluating the equilibrium points at the derivative and those are the eigenvalues or is there more to it? Thanks!

 

[mighty2000]

Great job on finding the equilibrium points and determining their stability! You are correct that the derivative at each equilibrium point can be considered as the eigenvalues for this equation. This is because the eigenvalues represent the rates of change at each equilibrium point, and in this case, the derivative at each point represents the rate of change of the function itself. So, you have already found the eigenvalues for this equation. However, if you were dealing with a system of equations, you would need to find the eigenvalues by setting up a matrix and solving for its eigenvalues. Keep up the good work!