Math46 – ODEs

Here are the course materials for my summer 2018 course on ordinary differential equations. If you have questions or comments, email me at .

Course Materials

Here are links to the syllabus, and to the quizzes and exams for the course.

Outline, Lecture Notes, and Homework

At the recommendation of Brandon Coya who taught this class the summer before, I’m following the presentation in Paul’s Online Notes on ODEs, which are easily accessible and provide a good exposition of the material and many examples. Here is an outline of the course, with links to the relevant sections in Paul’s Online Notes, along with my personal lecture notes (which I wrote for myself and may not be useful to others), and the homework that is due each week.

Week One – Terminology, and Separable and Homogeneous DEs

Lecture Notes · Homework
Talked about the syllabus and course logistics. Defined the basic stuff, like ODEs and PDEs, the solution to a differential equation, initial value problems, a general versus a particular solution to a DE, and the order of a differential equation. Also talked about directions fields, and how to find solutions to first-order separable and homogeneous differential equations.

Week Two – Exact, First-Order Linear, and Bernoulli DEs

Lecture Notes · Homework
Talked about using other substitutions, to solve some differential equations, and then about solving exact, first-order linear, and Bernoulli differential equations.

Week Three – Modelling and Equilibrium Solutions

Lecture Notes · Homework
Started talking about modelling physical situations with DEs, like modelling a falling body, and mixing problems and population growth. Also talked more about direction fields and equilibrium solutions of DEs, and how to interpret equilibrium solutions in terms of a physical situation we’re modelling.

Week Four – Nth-Order Linear DEs and Reduction of Order

Lecture Notes · Homework
Talked about linearly independent solutions to an Nth-order linear homogeneous DE and about the Wronksian. Did many examples of solving Nth-order linear homogeneous DEs with constant coefficients by looking at roots of their characteristic polynomials, and the cases where the roots are repeated or complex. Also talked about the method of reduction of order to solve second-order linear homogeneous DEs.

Week Five – Undetermined Coefficients, Variation of Parameters

Lecture Notes · Homework
Discussed non-homogeneous linear differential equations and how to solve them with the method of undetermined coefficients and variation of parameters.

Week Six – Laplace Transforms

Lecture Notes · Homework
Introduced Laplace transforms and all the details of their definition and then inverse Laplace transforms. Calculated a whole bunch of Laplace transforms manually before showing them the magic table of Laplace transforms. Then showed them how initial value problems can be easily solved with Laplace transforms. Also introduced the Heaviside function and how to use it to take the Laplace transform of piecewise-defined functions.

Week Seven – Systems of ODEs and More Modelling

Lecture Notes · Homework
First I needed to give the students a crash course on linear algebra: vectors, linear maps, determinants, and eigenvectors and eigenvalues. This was all to talk about systems of differential equations and how to solve an Nth-order differential equation by writing it as a system of first-order differential equations. Then I finished off talking about modelling with systems of differential equations, in particular modelling multiple tanks of intermixing water, and predator-prey models for population growth.

Thoughts and Reflections

Since my only exposure to differential equations before this course was taking a comparable class as an undergraduate, I tried to do a bunch of background reading to prepare to teach it. While I didn’t have anywhere near enough time to create the ideal course based on that preparation, I did come across a few key points to focus on if I teach it again. And hopefully listing these points here will help inspire other teachers who find themselves wanting to improve the infamous Introduction to ODEs course.

Give More “Solve this ODE” Exercises

A (surprising) complaint that a few students listed on their course evaluations was that there were too few ODE-solving drills on the homework. That’s an easy fix though: Just gotta grab a few of the popular ODEs textbooks, pillage some exercises, and take the time to work through those exercises myself to make sure that they are reasonable. I don’t think it would be useful to assign more drills as graded homework though (it can take quite a while to solve some ODEs after all), but it would be a cool to have a giant page of ODEs that can be solved using the techniques discussed to hand to the student towards the end of the course when they are thinking about the final. But while it’s important to provide students with many ODEs to solve to drill themselves, simply solving ODEs should not be the main emphasis of the course.

Incorporate the Advice in Professor Rota’s Essay

I remember ODEs class as being very computationally heavy and conceptually uninspiring, just presenting a laundry-list of techniques for solving different sorts of ODEs. Turns out these sentiments are not just angsty memories from my undergraduate years, but were validated by MIT mathematician Gian-Carlo Rota, in his essay Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations. Professor Rota basically decries the common syllabus for an standard Introduction to ODEs course as being outdated, and consisting largely as useless tricks to solve ODEs. While I disagree with some of Professor Rota’s points (I’ve gotta hope that the word problems I assign in the modelling sections, though contrived, do something to help the non-math majors in their non-math classes), I think that the points that Professor Rota makes are the best starting point to improving the current standard Intro to ODEs course.

Reading through the essay though, I realized I seriously lack the expertise to make any substantial changes to the standard laundry-list curriculum for the course. If I had more time to prepare I may have used a textbook that tries to adhere to Professor Rota’s lessons, like one of the books mentioned in this MathOverflow post. At the very least, even if I don’t use one of these textbooks it would be beneficial to learn the topics that a ODEs course can cover besides the usual laundry-list of ODE solving techniques.

Learn More About Differential Equations Myself

I study algebra, and prior to this course hadn’t thought about differential equations for years, which was why I was so unable to deviate from the standard curriculum for this course. Although it’d take some time, this course cannot be made better if the instructor doesn’t have the background to do so. Here’s a few things I could have learned more about, in descending order of importance, to have made the course better:

Second-order linear ODEs naturally model mechanically vibrations, given by systems of weights attached to springs. I skipped this section because of my lack of background in physics.
The method of variation of parameters for solving ODEs looks like magic, but the Wikipedia page for it suggests that some intuition for this method can be gained from thinking harder about the physical situation that linear ODE models.
I want to learn more about the phase plane, and phase spaces. This is the concept that the highly recommended textbook Ordinary Differential Equations Vladimir Arnol’d starts with.
Understand some of the points made by James Cook in his answer to this MathEdSE post.
I could relate the study of differential equations to my own research, and delve into the theory of $D$-modules.

Give Outside Reading and Resources to Students

It’s important that students realize that outside the course there are people thinking and learning the same stuff they are learning in the course. Basically they should have the class validated by the outside world. And what better way to do that than provide them with links to reading material and other resources to aid in their understanding. Here is a collection of a few such links:

Mike Pierce