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Preview
Preview
If you've ever jumped off a diving board or looked at an airplane wing during flight, you've seen cantilevers in action. These beams are engineered to be flexible but not too flexible. In this lesson we'll study a simplified model of how cantilevers bend. The model will turn out to be a fourth-order ODE. We'll then spend the rest of the lesson studying higher-order ODEs in general. We'll end up needing more facts about complex numbers and will also run into new challenges that will nudge us toward studying linear algebra, which we'll start doing in the next lesson.
In Module I we'll introduce the aforementioned fourth-order ODE cantilever dynamics model, in the context of aircraft design. We'll then discuss higher-order linear ODEs and the analogue of Theorem 3.2.1 (from Lesson 3), which describes the general solution of homogeneous nth-order ODEs (NOLDEs).
In Module II we'll specialize to constant-coefficient NOLDEs (CC-NOLDEs). Paralleling what we did in Lessons 4-5, we'll see that such ODEs have an associated nth-order characteristic equation, and we'll explore the real roots cases of it in this Module.
Finally, in Module III we'll explore the complex roots case of the characteristic equation of a CC-NOLDE, drawing on additional facts about complex numbers, including how to graph them, to help us solve CC-NOLDEs.
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 7.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 7 - Practice Problems.pdfPage updated
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