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This book originated from my class notes for teaching 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. I also taught Math 285 at UIUC and Math 20D at UCSD using this book. The standard book at UIUC was Edwards and Penney, Diﬀerential Equations and Boundary Value Problems: Computing and Modeling [EP], fourth edition. The standard book at UCSD is Boyce and DiPrima’s Elementary Diﬀerential Equations and Boundary Value Problems [BD]. As the syllabus at UIUC was based on [EP], the early chapters in the book have some resemblance to [EP] in choice of material and its sequence, and examples used. Among 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.

I taught the UIUC courses using the IODE software (http://www.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 http://www.jirka.org/genius.html). I used Genius in class to show these (and other) graphs.

This book is available from http://www.jirka.org/diffyqs/. Check there for any possible updates or errata. The LATEX source is also available from the same site for possible modiﬁcation and customization.

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, and probably others I have forgotten. Finally I would like to acknowledge NSF grants DMS-0900885 and DMS-1362337.

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 4 and 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 4 and 5. The textbook was originally done 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 § 4.10), chapter 5 (or 6 or 7 or 8).

Or a shorter version (Math 285 at UIUC) of the course at 3 hours a week for a semester:

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

The complete book can be covered in approximately 72 lectures, depending on the lecturers speed and not accounting for exams, review, or time spent in computer lab. Therefore, a two quarter course can easily be run with the material, and if one goes a bit slower than I do, then even a two semester course.

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.