Autonomous equations

Lesson 3M

An autonomous differential equation is one that depends only on \(y\) and \(y’\). It does not involve the dependent variable \(t\).

The phase line helps you understand these differential equations without solving.

Draw the phase line

Here is a video showing how to draw and interpret a phase line.

Practice

Find the equilibrium solutions of the differential equation \[\frac{dy}{dt} = (r-ay)y \] and determine if they are stable or unstable. How does the long term behavior of \(y\) depend on the initial value \(y_0=y(0)\)?

This is called a Logistic equation and has been used to model populations. In a population model, what is the meaning of the stable equilibrium?

Further reading

Read the beginning of Section 2.5, from the top of page 78 to the middle of page 81.

Looking for a challenge?

Solve the differential equation from the video: \[y’=(y-1)(y-2)(y-3).\] Use partial fractions to integrate.

If you are willing to use a page or two, you can even solve for \(y\) (using the quadratic formula) to get \[y(t)= 2 \pm \left[1-\left(1-\frac{1}{(y_0-2)^{2}}\right) e^{2t}\right]^{-1/2}. \]