Lesson 19: Classifying Equilibrium Points for Linear Systems
Preview
Can differential equations and linear algebra help us understand love? In this lesson we’ll explore that question via a mathematical model of the dynamics of romance between two individuals. The model we’ll study is a CC-S2FOLDE, a system of the form x-dot = Ax that we’ve studied in detail over the past few lessons. When what we know about solving such systems to our model we’ll discover that the patterns we noticed in the phase portraits in the previous lesson can be explained in terms of the eigenvalues of A, and that those patterns have real-world consequences for the fate of the relationship the system models. This application and the new connections it leads to among the various topics we’ve studied in the course make for an excellent ending point for the course.
We’ll begin in Module I by discussing the CC-S2FOLDE that models the love dynamics between two individuals. Then, we’ll study how to draw the phase portraits for linear plane autonomous systems, what we used to call x-dot = Ax systems.
In Module II we’ll explore the phase portraits of x-dot = Ax systems in more generality. We’ll discover that the eigenvalues of A control the qualitative features of the system’s phase portrait.
Finally, in Module III I won’t discuss new content but instead leave you with some parting thoughts about what we’ve studied and where you could go from here in your journey into higher mathematics.
Learn
Work through the lesson notes below, consulting the videos below it when you get to the "See Class Notes" boxes. For your records, the annotated lesson notes are below the videos. Some tips for you as you work through these resources:
I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.
Lesson Notes
Lesson 19.pdfClass Notes A
Class Notes B
Class Notes C
Class Notes D
Reflect
If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:
Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.
Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.
If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:
Please summarize the main mathematical takeaways from the lesson notes.
What was the most interesting part of what you learned, and why?
What, if anything, do you still find confusing?
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood.
Lesson 19 - Practice Problems.pdf