Notes on Diffy Qs: Differential Equations for Engineers

By: Jiří Lebl
(website #1 (personal), website #2 (work: OSU), email: )

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A one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence. This free online book (e-book in webspeak) should be usable as a stand-alone textbook or as a companion to a course using another book such as Edwards and Penney, Differential Equations and Boundary Value Problems: Computing and Modeling or Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems (section correspondence to these two is given). I developed and used this book to teach Math 286/285 at the University of Illinois at Urbana-Champaign Sample Dirichlet problem solution (one is a 4-day-a-week, the other a 3-day-a-week semester-long course). I also taught Math 20D at University of California, San Diego with this book (a 3-day-a-week quarter-long course). There is enough material to run a 2-quarter course, and even perhaps a two semester course depending on lecturer speed.

The aim is to provide a low cost, redistributable, not overly long, high quality textbook that students will keep rather than selling back after the semester is over. Even if the students throw it out, they can always look it up on the net again. You are free to have a local bookstore or copy store make and sell copies for your students. See below about the license.

Another reason is to allow modification and customization for a specific purpose if necessary. If you do modify the book, make sure to mark it prominently as such to avoid confusion. This aspect is also important for longevity of the book. The book can be updated and modified even if I happen to drop off the face of the earth. You do not have to depend on any publisher being interested as with traditional textbooks. Furthermore, errata are fixed promptly, meaning simply that if you teach the same class next term, all errata that are spotted are already fixed. No need to wait several years for a new edition.

The graphs in the book were created using the Genius software.

MAA published a review of the book (they looked at the December 2012 edition). Temperatures of object with ambient temperature oscillating

Table of contents:

1. First order ODEs
2. Higher order linear ODEs
3. Systems of ODEs
4. Fourier series and PDEs
5. Eigenvalue problems
6. The Laplace transform
7. Power series methods
8. Nonlinear systems (new, Oct/20/2014)

There are currently 623 exercises throughout the book (March 21st 2017 edition), 201 of which have a solution in the back (those numbered 101 and above). A few exercises are within the section text, but most are in their own subsection at the end of every section. Each section should have enough exercises for homework even for a demanding class.

Please let me know at if you find any typos or have corrections, extra exercises or material, or any other comments. I will always keep all older versions available for download, at least when there are nontrivial updates. When the updates are reasonably minor, I will try to preserve pagination and numbering of sections/examples/theorems/equations/exercises as much as possible.


Do let me know () if you use the book for teaching a course! The book was used, or is being used (other than my courses at UIUC and UCSD), at over a dozen universities including Dartmouth College, University of Tennessee, University of Toledo, University of British Columbia, University of California at Irvine, University of Kentucky, University of Hawaii, and many others. The Saylor Foundation is using it as one of the books for their online Math 221 course.

The book has been selected as an Approved Textbook in the American Institute of Mathematics Open Textbook Initiative.

See a list of classroom adoptions for more details.


Download the book as PDF
(March 21st, 2017, version 5.1, 371 pages, approximately 3.2 MB download)

Version 5.0 added section 0.3 (classification) and section 1.8 (exact equations)

Look at the errata in the current revision (if any).

Look at the change log to see what changed in the latest version (there you can also download old versions if you wish).

Buy paperback:

I get a bit of money when you buy these (depending on where exactly they are bought, most if it is bought at the createspace store). Probably enough to buy me a coffee, so by buying a copy you will support this project. You will also save your toner cartridge. The difference between these two versions is essentially just the cover art. I have seen printed versions from both and they are both good quality.

Differential Equations Cover
Buy a paperback copy at for $15.68. You can also buy a larger (letter size) coil-bound copy at for $18.28. This copy is the March 21st, 2017 (version 5.1) revision. No ISBN for this guy.

Differential Equations Cover
Buy a copy at for $14.90, or buy a copy on Amazon. This should be a little easier to buy and quicker/cheaper to ship than lulu. This copy is the March 21st 2017 (version 5.1). This one has an ISBN:
ISBN-13: 978-1541329058
ISBN-10: 1541329058

(The older October 20th, 2014 revision is also up on amazon with ISBN-13: 978-1505698190, if you really need the old version)

Web version:

Browse the HTML version of the book (for easier reading on the web). The PDF version is still the canonical version however. Some things may look strange or may be hard to read in the web version simply because of imperfections in the conversion.

Interactive SAGE demos:

Section 1.6: Several interactive demos on autonomous equations in one variable.

Section 1.7: An interactive demo of Euler's method.

Section 2.4: Several animations of mechanical vibrations.

Section 2.6: Interactive demo of forced oscillations and resonance.

Section 3.5: Interactive demos of two-dimensional autonomous systems.

Section 3.6: A second order system (two carts with springs between them) interactive demo.

Instructor resources:

I put together all the figures as PDFs as one big zipfile. This should make it easier to create computer slides using the figures if you want without messing with the source tarball below. If a figure appears in multiple places, its filename only refers to the first such place.

There's tons of extra materials (including longer modeling projects) at SIMIODE.

Translations and derivative versions:

Prof. Martin Weilandt of Universidade de Santa Catarina has prepared a partial Portuguese translation. See his class page.

Prof. Charles Bergeron has created a modified version of the book based heavily on Notes on Diffy Qs. The title is Differential Equations: Including Linear Algebra Topics And Computer-Aided Problem-Solving. The book removes some topics (e.g. PDEs), but adds an entire linear algebra chapter. Also the book covers the use of the computer algebra system Maxima in the context of the material.


LaTeX source as a tarball. The main file is diffyqs.tex. I compile the pdf with pdflatex. You also want to run makeindex to generate the index (I generally run pdflatex diffyqs three times, then makeindex diffyqs, and then finally pdflatex diffyqs again). The setup file with all the preamble you may want to edit is diffyqssetup.sty.

During the writing of this book, the author was in part supported by NSF grant DMS-0900885 and DMS-1362337.


Creative Commons License Creative Commons License This work is dual licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License and Creative Commons Attribution-Share Alike 4.0 License. You can use, print, copy, and share this book as much as you want. You can base your own book/notes on these and reuse parts if you keep the license the same (that is, as long as you use at least one of the two licenses).

Integrating factor formula

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