Lesson 15: Solving Systems of First-Order ODEs
Preview
In Lesson 4 we discussed the mass-spring-dashpot model and how automotive engineers use it to smooth out car rides over bumpy roads. It turns out that the same system can be used to model the tiny oscillations atoms undergo within a molecule. In the simplest cases, the mathematical model that results is a variant of the system x-dot = Ax that we’ve been studying. This sister system can be solved using eigenvalues and eigenvectors, and the results turn out to have physically meaningful interpretations. We’ll discuss those interpretations in the practice problems. But first we’ll talk about how to solve x-dot = Ax using just eigenvalues and eigenvectors.
In Module I we’ll learn how to solve x-dot = Ax using eigenvalues and eigenvectors. But we’ll need to discuss a couple of more theorems before we can assemble the solutions we find into the general solution.
Finally, In Module II we’ll discuss those additional theorems and develop a procedure for finding the general solution to x-dot = Ax.
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 15.pdfClass Notes A
Class Notes B
Class Notes C
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 15 - Practice Problems.pdf