# Differential Equations Notes

## Goal

The goal of these differential equations notes is to eventually cover all standard topics in a typical two-semester course sequence in differential equations at the undergraduate level (such courses are often taken by math, engineering, and science majors).

## Prerequisites

Prerequisites include trigonometry, the differential calculus (derivatives), and the integral calculus (integrals) through integration techniques such as substitution and integration by parts. We will make use of some power series, multivariate calculus, linear algebra, and complex variables, but the necessary concepts will be explained as needed.

## Usage Tips

These differential equations notes consist of a series of “digital notecards,” each accompanied by a video explanation. There are at least two ways to use these notes.

- You can use the notecards to rapidly review what you’ve already learned, watching the videos only when you cannot provide the correct answer to a notecard’s prompt.
- You can learn core topics by watching the videos, and then you can quiz yourself by attempting to answer the prompt on the corresponding notecard. You can do this to learn on your own (in which case I recommend working additional exercises for practice), or to supplement a university course. In the latter case, I recommend watching the videos before your lecture if possible. That way, the lecture will serve as a review session, and you’ll be able to follow it much better!

## Course Outline (with links to notes)

Module: Introduction to Differential Equations

### Lesson: Examples, Models, and Slope Fields

- Terminology: What is a differential equation?
- Connection: Differential Equations vs. Algebraic Equations
- Example: How to Check a Solution of a Differential Equation
- Example: How to Make and Interpret a Basic Slope Field
- Example: How to Solve an Initial Value Problem (Exponential Growth)
- Terminology: General Solutions vs. Particular Solutions

## References

Each of the following resources covers most of the topics that will be included in the differential equations notes presented on this site. The first three books have been adopted by many universities for their differential equations courses. However, this is just a sample; there are many other books with similar coverage.

- Elementary Differential Equations and Boundary Value Problems by William E. Boyce and Richard C. DiPrima (the link is to the older eighth edition, since as I write this, a used copy of the eighth edition is only about $5 with shipping from Amazon, and the coverage is very similar to that of the current edition)
- Elementary Differential Equations with Boundary Value Problems by C. Henry Edwards and David E. Penney (this link is to the older fifth edition, which is less expensive than the current edition)
- Differential Equations with Boundary-Value Problems by Dennis G. Zill and Michael R. Cullen (this link is to the older 6th edition, which is much cheaper; this text is now at least in its 8th edition with authors Zill and Wright)
- You can also find free textbooks on differential equations and other math subjects that have been approved by the American Institute of Mathematics.
- Paul’s Online Math Notes (Differential Equations Notes) are a popular, and free, text-based resource.