Lesson 3: Second-Order Linear ODEs
Preview
What is the curve a ball thrown up in the air follows as it moves? It turns out that the answer to this question involves second-order differential equations. We will begin our study of these ODEs in this lesson and later apply what we learn to determine the answer to the question above.
In Module I we’ll derive the second-order ODEs (SODEs) modeling projectile motion on Earth. We will then broaden our sights to study the theory behind second-order ODEs (SODEs), specializing to the linear cases (SOLDEs).
In Module II we’ll discuss the special structure of solutions to homogeneous SOLDEs. We will learn that some such solutions are special, in that every other solution to a homogeneous SOLDE can be expressed as a combination of these special solutions. This result will launch us on the hunt for "special" solutions to homogeneous SOLDEs, a search we will conduct over the next few lessons.
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 3.pdfClass Notes A
Class Notes B
Class Notes C
Class Notes D
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 3 - Practice Problems.pdf