Differential Equations Books - Best Textbooks for Differential Equations

In graduate school, I was taking a course in PDE's (the multivariable form of equations, following ODE's) that I found both tedious and difficult, and one night I was stuck on the second problem in a large problem set. After wrestling with the math for a few hours, I gave up and decided to turn in for the night by having a couple beers.

After drinking two beers rather quickly and feeling a little tipsy, I had the strange urge to return to my problem set. I tried something new, got un-stuck, and not only finished the problem I was working on, but I finished the remainder of the problems--and I did so quickly, in less time than I had been working on the set to begin with.

The next day I got up early to check my work--could I really trust myself to do PDE's after two beers? Apparently, I could. My work was perfect, and I handed the problem set in and got a perfect score. What's the moral of this story?

Differential equations, especially PDE's, can be intimidating. But it's not a hard subject the way other areas of math like abstract algebra are. I am inherently better at abstract algebra than PDE's, but I can't do it drunk. The problem with ODE's and PDE's is usually that there are a lot of tedious manipulations, and that some problems are just so intimidating-looking that you can get frazzled and not know where to get started.

Loosen up a bit. You will find it's a lot easier than you think. You're probably the #1 person holding yourself back! If you tell yourself that it's tough, then it'll be tough. It's not that bad! You want bad? Try real analysis or abstract algebra, and make sure you don't have too much to drink.

Before you buy a specific book (or for the teachers and professors, choose a textbook for your course), I think it is important to ask yourself a question about what type of book you're looking for. I have divided the books here into three categories:

• Mainstream introductory texts - These are the books that most students know as math books. They are big, expensive (unless you buy older editions), and have pretty covers. Most have been through many editions. These books tend to be more accessible and have more explanations, although some are better than others. I think these books are good for students who are learning the subject for the first time, and in some cases, they are the only books that make sense for students who are not going to be studying further math. But they're not the best books for all students.
• Application-oriented books - Diff-EQ, more so than any other area of mathematics, is a field that is almost totally driven by applications. It is essential in areas as diverse as engineering, physics, chemistry, and economics. The field as a whole is driven by problems that arise in the real-world more so than many other areas of calculus or mathematics. I think it can be very illuminating to learn the subject through books that discuss how and why the equations arise in the natural world. I have included several of these books here as well.
• More theoretical books - I include one particularly advanced book on PDE's, but I also include two "yellow books" that bridge the gap between theoretical books and application-oriented books. Although these books may strike some students as more advanced and tougher to understand, I think their conciseness can also make the subject more accessible and less intimidating to some students. I particularly recommend checking out the two yellow books in the Springer series, reviewed below.

Boyce and DiPrima's book is my favorite of the mainstream textbooks in differential equations. It is accessible and clear, yet covers more advanced material than is typical.

Pros: I find the Boyce and DiPrima to be an outstanding for a course textbook, for self-study, and as a reference. The presentation is clear, the order in which the topics are presented feels natural to me, and I think this book does a better job than most of showing how the various techniques are related to each other.

Cons: This book suffers from what I call edition creep: the publishers have released far more edition than are justified, in my opinion. I recommend buying an older edition. I've worked with 5th, 6th, 7th, and 8th editions and all of them were excellent. In my opinion, there is no reason to order a newer edition unless you have a specific need for it (like a course which is pulling homework problems from a newer edition).

This is a little-known volume that I think is remarkable effective at presenting this subject as well as the closely related subject of dynamical systems. An overwhelming majority of ODE textbooks follow more or less the same format. This book is quite different from these generic textbooks, and may offer a refreshing change of pace for people who are not satisfied with the standard presentation of this subject.

Pros: Very well-written, and surprisingly accessible. Requires little background other than introductory calculus. The authors strongly emphasize intuition and a qualitative approach, although they also discuss analytic and numerical approaches as well. I like the presentation of the subject and the choice of topics covered much more in this book than in the normal intro texts. I also find it is more practical for people going into applied math, because the presentation of the subject is less idealized and oversimplified.

Cons: More concise than typical intro texts, and has a rather different development of the subject from what is typical in most college curricula. Some students may find this book needs to be supplemented by a more mainstream textbook.

I find this little yellow volume by Martin Braun to be absolutely delightful; it is very different from the mainstream texts in the subject, but I like its presentation much better.

Pros: Lucid prose. Does a better job of explaining how and why various techniques for solving certain types of equations are applicable in different situations. Separates application and theory into different sections so that a student (or teacher) can focus on one without getting bogged down in the other, but interweaves them enough that this book is very useful for helping students to see and work with applications of the theory. Gets into numerical approaches too, although starts first with analytic techniques and presents numerical approaches later, which I find helpful.

Cons: The use of programming in this text is a bit antiquated. When I read this book, the Pascal and FORTRAN programs were shown in the text and the C programs relegated to the index; I'd prefer the other way around. Some students may find this book slightly too concise.

Partial differential equations or PDE's are the multi-variable form of differential equations, and are thus a more advanced subject. This is the most accessible introductory textbook on this subject that I've been able to find.

Pros: PDE's are tough, and this book makes them about as easy as possible. There is rich discussion, and the author, Richard Haberman, provides thorough motivation for each technique. I find this book is a great first introduction to PDE's, for a student who has a good understanding of differential equations in a single variable but no prior exposure to PDE's.

Cons: I found the presentation in this text to be overly idealized at times. PDE's in real world applications can be quite ugly. I think this book is complemented by the book Applied Mathematics by Logan, which is more practical than idealized, and addresses the weaknesses in this text. If you need to go deeper in theory, especially dealing with non-linear equations, I recommend the Evans text. I review both of these below.

Above I gave a review of an introductory textbook in the topic of PDE's. This volume covers the same topic but for a more advanced audience.

Pros: This book is quite deep. It is also quite modern, covering a lot of newer techniques that are omitted from more classical textbooks. There is a particular emphasis on non-linear equations, which may be refreshing as non-linear equations are quite ubiquitous in applications, and often glossed over or outright ignored in introductory courses and textbooks. The typography is neat, typical for the AMS series it is a part of.

Cons: Probably too advanced for most students seeing this subject for the first time. It is also highly abstract and theoretical, and does not go into discussion about applications or practical considerations. It is probably of most use for theoretical researchers who are working in the field of PDE's. It will probably be less useful to scientists and engineers who encounter these sorts of systems in their work.

I attended my first round of graduate school at the University of Delaware, where I majored in what was called "Applied Mathematics", getting a Master's of Science degree in 2007. At Delaware, in the department I was in, Applied Math basically meant differential equations: there was an older group of professors who tackled problems primarily using analytical methods, and a newer school of mostly-younger professors who were interested in numerical methods.

Although I loved many aspects of the program at Delaware, and learned a tremendous amount while being there, I thought this view of applied mathematics was a bit narrow. When I've talked to people working and researching at other universities I've realized that there are numerous other branches of mathematics that can be useful in applications in science, engineering, social sciences, business, and many other fields. These even include "weird" or "esoteric" branches of math like algebraic topology or combinatorics, which at Delaware were lumped under the "pure math" heading.

I encourage people to think broadly...differential equations can be applied to many aspects of life, but they're not the only branch of mathematics that falls under the "applied math" heading, and I don't think it's particularly truthful or constructive to equate the two the way some math departments do.

J. David Logan's Applied Mathematics is not strictly a differential equations textbook, but it does dedicate a significant portion of the text to the subject of differential equations, and it is also closely related.

Pros: Easy to skip around in. Makes a great reference text. Philosophical and deep, subtle and thought-provoking. I find that this book prepares the reader to actually use mathematical techniques in practical applications more than most other textbooks.

Cons: This book is perhaps misnamed, and is much narrower than the name might lead you to believe; it is really not about applied mathematics as a whole, so much as it is about specific fields within applied mathematics. The book focuses on deterministic modeling, covering such fields as PDE's, the calculus of variations, perturbation theory. There is minimal discussion of probability, and not so much as a mention of statistics or combinatorics, two fields I consider essential parts of applied mathematics.

Although there are not as many true textbooks in the area of ODE's and PDE's available for free download, there are quite a few good free lecture notes. Here are three that I hope will be most useful:

• Notes on Diffy Qs: Differential Equations for Engineers - This collection of lecture notes can be used as a standalone textbook for self-study, or as a supplement. It is maintained by Jiri Lebl, a mathematics professor at the University of Wisconsin Madison, and hosted on his personal website.
• Notes on Differential Equations (PDF Link) - by Bob Terrell, hosted on Cornell's website. Bob Terrell is a Cornell professor who is experienced teaching this subject and many other related subjects in mathematics, including Linear Algebra and Multivariable Calculus. His research is in fluid mechanics, one of the applications of diff-eq.
• Linear Partial Differential Equations and Fourier Theory (PDF Link) - by Marcus Pivato, Trent University, Canada. Marcus Pivato is a professor at Trent University and researches economics, ergodic theory, and other dynamical systems.