Lesson 2: Separable ODEs
Preview
Some ancient cultures used "water clocks" to measure the passage of time. In these devices, water drains at a predictable rate from a tank with a hole in it, and the water level in the tank is associated with particular time stamps (e.g., 1 hour, 2 hours, etc.). In this lesson we'll develop a mathematical model of a simple water clock. It will turn out to be a non-linear FODE, which will motivate us to study one family of solvable non-linear FODEs: separable FODEs.
In Module I we'll derive the separable FODE associated with the water clock model and learn how to solve simple separable FODEs.
In Module II we'll discuss more advanced solution techniques for separable FODEs, including the usage of partial fractions. We will also discuss additional applications of separable FODEs.
Review
Short differential calculus refresher (scroll down to the "Complete Calculus Cheat Sheet")
Implicit differentiation (here is a short video review of that concept, from Khan Academy)
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 2.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 2 - Practice Problems.pdf