Lesson 14: Systems of First-Order ODEs
Preview
While many real-world phenomena can be accurately modeled by a single ODE—as we have done many times throughout this course—some phenomena require two or more ODEs to model. In such settings one obtains a system of differential equations that describe the evolution of the system (generally the evolution over time); we call these dynamical systems. In this lesson we’ll lay the foundations for analyzing dynamical systems.
In Module I we’ll discuss a familiar and important example of a dynamical system: a hurricane. We’ll briefly discuss how it can be modeled—in a drastically simplified way—by a system of first-order differential equations. We’ll spend the remainder of the module studying how such systems arise from the NODEs we’ve been studying in the course.
In Module II we’ll define a general system of first-order ODEs and introduce new terminology to connect that new concept back to CC-NOLDEs.
Finally, in Module III we’ll start using the linear algebra we learned in Unit 3 to help us understand the structure of the solution sets to systems of first-order ODEs.
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 14.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 14 - Practice Problems.pdf