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Preview
Preview
We learned about direction fields in the previous lesson and how they help us visualize the solutions to plane autonomous systems without having to actually solve those systems. Specifically, by “following the arrows” in a direction field we get a visual of what the orbits look like for the system. But we can we calculate the equation of those orbits, or must we settle for just getting a rough idea of what the orbits look like by relying in numerical calculations? In this lesson we’ll learn about one way to calculate the orbits of a plane autonomous system. Before launching into that, I’ll motivate our work in this lesson by considering a well-known model of predator-prey dynamics that although we don’t know the explicit solutions to we do know the equation of the orbits to. Those orbit equations turn out to tell us everything we need to know about the dynamics without needing the solution to the system.
We’ll begin in Module I by calculating the orbits for a simple-enough system by using the solutions for the system.
In Module II we’ll discuss an additional method that we can use which doesn’t require having solved the system. As we’ll see, this method will bring us back to Unit 1, specifically, to solving first-order ODEs.
Finally, in Module III we’ll complete our analysis of orbits by learning how to identify the correct direction of the orbits.
Review
Review
Learn
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 Notes
Lesson 18.pdfClass Notes A
Class Notes A
Class Notes B
Class Notes B
Class Notes C
Class Notes C
Class Notes D
Class Notes D
Class Notes E
Class Notes E
Reflect
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
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood.
Lesson 18 - Practice Problems.pdfPage updated
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