## Section0.1Notes about these notes

¶This book originated from my class notes for Math 286 at the University of Illinois at Urbana-Champaign in Fall 2008 and Spring 2009. It is a first course on differential equations for engineers. Using this book, I also taught Math 285 at UIUC, Math 20D at UCSD, and Math 4233 at OSU. Normally these courses are taught with Edwards and Penney, *Differential Equations and Boundary Value Problems: Computing and Modeling* [EP], or Boyce and DiPrima's *Elementary Differential Equations and Boundary Value Problems* [BD], and this book aims to be more or less a drop-in replacement. Other books I used as sources of information and inspiration are E.L. Ince's classic (and inexpensive) *Ordinary Differential Equations* [I], Stanley Farlow's *Differential Equations and Their Applications* [F], now available from Dover, Berg and McGregor's *Elementary Partial Differential Equations* [BM], and William Trench's free book *Elementary Differential Equations with Boundary Value Problems* [T]. See the Further Reading chapter at the end of the book.

The organization of this book to some degree requires chapters be done in order. Later chapters can be dropped. The dependence of the material covered is roughly:

There are some references in chapters Chapter 4 and Chapter 5 to material from Chapter 3 (some linear algebra), but these references are not absolutely essential and can be skimmed over, so Chapter 3 can safely be dropped, while still covering chapters Chapter 4 and Chapter 5. The textbook was originally made for two types of courses. Either at 4 hours a week for a semester (Math 286 at UIUC):

Introduction, Chapter 1, Chapter 2, Chapter 3, Chapter 4 (w/o Section 4.10), Chapter 5 (or Chapter 6 or Chapter 7 or Chapter 8).

For a shorter version (3 hours a week) of the above course, skip chapters Chapter 4 and Chapter 5. The second type of the course at 3 hours a week (Math 285 at UIUC) was:

Introduction, Chapter 1, Chapter 2, Chapter 4 (w/o Section 4.10), (and maybe Chapter 5, Chapter 6, or Chapter 7).

The complete book can be covered at a reasonably fast pace at approximately 75 lectures, not accounting for exams, review, or time spent in computer lab. A two quarter course can be easily run with the material, and if one does not rush, a two semester course. For example:

Semester 1: Introduction, Chapter 1, Chapter 2, Chapter 6, Chapter 7, (and perhaps bits of Chapter 4).

Semester 2: Chapter 3, Chapter 8, Chapter 4, Chapter 5.

The chapter on Laplace transform (Chapter 6), the chapter on Sturm-Liouville (Chapter 5), the chapter on power series (Chapter 7), and the chapter on nonlinear systems (Chapter 8), are more or less interchangeable time-wise. If Chapter 8 is covered it may be best to place it right after Chapter 3. If time is short, the first two sections of Chapter 7 make a reasonable self-contained unit.

I taught the UIUC courses using the IODE software (https://faculty.math.uiuc.edu/iode/). IODE is a free software package that works with Matlab (proprietary) or Octave (free software). Unfortunately IODE is not kept up to date at this point, and may have trouble running on newer versions of Matlab. The graphs in the book were made with the Genius software (see https://www.jirka.org/genius.html). I used Genius in class to show these (and other) graphs.

This book is available from https://www.jirka.org/diffyqs/. Check there for any possible updates or errata. The LaTeX source is also available for possible modification and customization.

*Acknowlegements:* Firstly, I would like to acknowledge Rick Laugesen. I used his handwritten class notes the first time I taught Math 286. My organization of this book through chapter 5, and the choice of material covered, is heavily influenced by his notes. Many examples and computations are taken from his notes. I am also heavily indebted to Rick for all the advice he has given me, not just on teaching Math 286. For spotting errors and other suggestions, I would also like to acknowledge (in no particular order): John P. D'Angelo, Sean Raleigh, Jessica Robinson, Michael Angelini, Leonardo Gomes, Jeff Winegar, Ian Simon, Thomas Wicklund, Eliot Brenner, Sean Robinson, Jannett Susberry, Dana Al-Quadi, Cesar Alvarez, Cem Bagdatlioglu, Nathan Wong, Alison Shive, Shawn White, Wing Yip Ho, Joanne Shin, Gladys Cruz, Jonathan Gomez, Janelle Louie, Navid Froutan, Grace Victorine, Paul Pearson, Jared Teague, Ziad Adwan, Martin Weilandt, Sönmez Şahutoğlu, Pete Peterson, Thomas Gresham, Prentiss Hyde, Jai Welch, Simon Tse, Andrew Browning, James Choi, Dusty Grundmeier, John Marriott, Jim Kruidenier, Barry Conrad, Wesley Snider, Colton Koop, Sarah Morse, Erik Boczko, Asif Shakeel, Chris Peterson, Nicholas Hu, Paul Seeburger, Jonathan McCormick, and probably others I have forgotten. Finally I would like to acknowledge NSF grants DMS-0900885 and DMS-1362337.