Lesson 17: The Qualitative Study of ODEs
Preview
Our foray into studying x-dot = Ax was motivated in part by the dynamical systems tie-in that we discussed in Module I of Lesson 14. Now that we’ve learned how to solve x-dot = Ax, we will start using our results to visualize our solutions in ways that generate additional insights. This approach will explain, for example, why the “wind field” in Figure 14.1(b) is indeed the “wind field” associated with the system (14.1) in Lesson 14. Furthermore, as we will soon see, the approach to visualizing solutions to SNFODEs we will build up to—what is known as the qualitative theory of ODEs—will enable us to glean properties of the solution to the SNFODE without having to solve the system.
We will being in Module I by considering more general SNFODEs, namely ones that are not necessarily linear. We’ll then define the notion of an equilibrium solution and then explore what these look like in the real world via an applied example. for the most part, specialize our study S2FODEs, since these are the easiest to visualize.
In Module II we’ll specialize our study to S2FODEs, what we’ll call planar systems, since these are the easiest to visualize. We’ll introduce a method to visualize such systems called the direction field that does not require that we calculate the solution to the system first.
Finally, in Module III we’ll introduce the notion of an orbit and discuss how that helps us visualize the solutions to planar systems without needing to solve the system.
Review
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 17.pdfClass Notes A
Class Notes B
Class Notes C
Class Notes D
Class Notes E
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 17 - Practice Problems.pdf