Good differential equations text for undergraduates who want to become pure mathematicians

Question

Alright, so I have been taking a while to soak in as much advanced mathematics as an undergraduate as possible, taking courses in algebra, topology, complex analysis (a less rigorous undergraduate version of the usual graduate course at my university), analysis, model theory, and number theory. That is, I have taken enough 'abstract' (proof-based) mathematics courses to fall in love with the subject and decide to pursue it as a career.

However, I have been putting off taking a required ordinary differential equations course (colloquially referred to as 'calc 4', though this seems inappropriate) which will likely be very computational and designed to cater to the overpopulation of engineering students at my university.

So my question is, for someone who might have to actually concern themselves with the theory behind the 'rules' and theorems which will likely go unproven in this low-level course (likely of questionable mathematical content), what might be a decent supplementary text in ODE? That is, something substantive to counter-balance the 'ODE for students of science and engineering'-type text I will have to wade through. I want to study algebraic geometry further (I have gone through Karen Smith's text and the first part of Hartshorne), so something which goes from basic material through differential forms and related material would be nice.

Thanks! (and yes, it's embarrassing that I still haven't taken the 200-level ODE course, but I have been putting it off in favor of more interesting/rigorous courses... but now there's that whole graduation requirements issue). --Lambdafunctor

Answer 1

Maybe I am reading too much into your pseudonym and your partly apologetic and partly condescending comments about the course you are going to take, but please,

Don't disparage the "rules" and computational aspects of differential equations.

Firstly, it is a beautiful subject with direct scientific origin and arguably most applications (save only calculus, perhaps) of all the courses you'd ever take. Secondly, these scientific connections continue to motivate and shape the development of the subject. Thirdly, rigor and abstraction are not substitutes for the actual mathematical content. Bourbaki never wrote a volume on differential equations, and the reason, I think, is that the subject is too content-rich to be amenable to axiomatic treatment. Finally, I've taught students who were gung-ho about rigorous real analysis, Rudin style, but couldn't compute the Taylor expansion of $\sqrt{1+x^3}.$ Knowing that the Riemann-Hilbert correspondence is an equivalence of triangulated categories may feel empowering, but as a matter of technique, it is mere stardust compared with the power of being able to compute the monodromy of a Fuchsian differential equation by hand.

Having forewarned you, here are my favorite introductory books on differential equations, all eminently suitable for self-study:

• Piskunov, Differential and integral calculus
• Filippov, Problems in differential equations
• Arnold, Ordinary differential equations
• Poincaré, On curves defined by differential equations
• Arnold, Geometric theory of differential equations
• Arnold, Mathematical methods of classical mechanics

You will find a lot of geometry, including an excellent exposition of calculus on manifolds, in the right context, in Arnold's Mathematical methods.

Answer 2

1. Arnol'd's ODEs.

2. Hirsch and Smale. As a second best the `supersized version' of this with Devaney added as a co-author.

Answer 3

You're in luck, lambda-since within the last few years,quite a few excellent advanced ODE texts have been published, in addition to the standard treatises. First, the more standard texts. If you want a strong theoretical course in ODE's, you really need to decide how strong you want it. A full theoretical presentation requires functional analysis and graduate real variables. I don't think you want anything that advanced,a t least not yet. So I'll recommend some of the best "intermediate" level texts - they're the most enjoyable to read, anyway.

My favorite is the beautiful geometric text Ordinary Differential Equations by Vladimir Arnold, in its' third (and sadly final) edition. Not only does it contain a rigorous exposition of ODE's and dynamical systems on manifolds, it contains a wealth of applications to physics,primarily classical mechanics. You'll need a strong background in theoretical calculus and linear algebra to read this one. So worth it.

A book I found immensely helpful when learning this material was Lawrence Perko's Differemtial Equations and Dynamical Systems. Not only does it cover more than Arnold's book, particularly on dynamical systems and nonlinear ODE's, it has a wealth of excellent exercises and diagrams of integral curves in a multitude of solution spaces/dynamical aystems,so important when learning the subject.

The old classic by Smale and Hirsch,Differential Equations,Dynamical Systems and Linear Algebra is best balanced by the second edition coauthored with Robert Devaney, Differential Equations,Dynamical Systems and An Introduction To Chaos. The second edition is more applied and less mathematically rigorous,but it contains much more information on nonlinear ODEs and chaotic dynamical systems. It also has many more pictures which are quite helpful in this subject-the sheer complexity of nonlinear systems really makes learning them nongeometrically strikingly noninformative. I would strongly advice getting BOTH books(the first edition is very pricey; I'd recommend borrowing it) and using thier union. Thier union may be the single best textbook that currently exists on the subject.

Lastly, there's James D.Miess' Differential Dynamical Systems, which contains not only a slightly more advanced presentation of the same material as Arnold and Perko, it contains many more applications and computer programming implementations,mainly to chemistry and classical mechanics.

All these books are outstanding and I think you'll find what you're looking for among them.

Answer 4

If you don't mind considering a recommendation from one of the co-authors of an ODE textbook, you sound like just the sort of student that we had in mind when we wrote "Differential Equations, Mechanics, and Computation". There is a Companion Website for our text at "http://vmm.math.uci.edu/ODEandCM/ where you will can find freely downloadable pdf files of more that half the book, including the entire introductory section, from which you can judge whether you want to use this as your introduction to ODE. A major consideration in writing the book was that it should be "easy" to read for a dedicated student looking for a conceptual introduction to the subject. The book was published by The American Mathematical Society in December 2009, and was reviewed the Mathematical Assoc. of America here: http://www.maa.org/maa_reviews/0211102.html Good luck with learning a truly beautiful subject, and my hope our book helps you. Richard Palais

Answer 5

2 years later, I understand this answer might not be helpful to you, lamdbafunctor, but for all of the other undergrads who come here and will see this, I believe Boyce and DiPrima's "Elementary Differential Equations and Boundary Value Problems" is exactly what you are looking for. The initial 4 chapter sequence this book follows (First order linear and nonlinear -> Second order linear and nonlinear -> Higher order linear and nonlinear) allows you to see the basic fundamentals being extended to more and more general cases and with very terse yet thorough and meaningful explanations through the entire way, it was a joy to read. From what I saw, it was almost like the book was written explicitly for self-study, as there is very little assumed detail. Many engineers find the downside to this book to be the almost complete lack of real-world modeling examples and such, and my response to them is that the purpose of Boyce/DiPrima is to gain a firm grounding in theory, while the purpose of other books like Edwards/Penney is to gain a firm grounding in physical/real-world applications. I am currently finishing up my first semester in Honors Diff Eq sophomore year, and I owe it almost entirely to this book.

If it lends any credibility to the argument, KhanAcademy's Sal Khan mentioned this was the book he was taught from when he learned diff eq's at MIT, and his selection of videos compliment this book perfectly: http://www.khanacademy.org/math/differential-equations

Answer 6

I personally like Jack Hale's book titled "Ordinary Differential Equations". I am a 3rd year graduate student studying Hamiltonian systems and have needed to review/learn topics in ODE's from time to time, and Hale's book has stood out as the most valuable resource for me.

I found Hale's book to be most readable, well-organized, and informative book covering the basics of ODE's. The treatment of the most basic issue of ODE's, i.e. existence/uniquness, is extremely well-written.

A lot of people seem to like Arnold's ODE book, and although it is a good book, I've had much better luck learning from Hale's book. Except for introducing differential equations on manifolds, all the main topics in Arnold's book are a subset of those in Hale's book. Hale also covers topics such as the Poincare-Bendixson Theorem and gets into stable/unstable manifolds, neither of which are present in Arnold's book. The presentation on time-periodic systems and related stability issues is also much clearer in Hale's book.

Another book I occasionally look at is Smale and Hirsh's book. This book is much more elementary than Hales and Arnold's, but has a few nice examples, especially in the few chapters regarding applications. However, the early section on The Poincare map is horrible, and should be the last place a person goes to learn what the Poincare map is.

To recap, if I need to look something up I looked at:

Hale
Arnold
Smale and Hirsch

Every time Hale's book wins in terms of readability, depth, and being easy to navigate through.

Last, but not least: Hale's book is published by Dover, and is quite cheap.

Answer 7

Just want to chime in with another book that hasn't been mentioned so-far: Philip Hartman's Ordinary Differential Equations. I am slightly ashamed to say that I still haven't read all of it, but it is the one that is on my shelf that I reach for if I need to look something up about ODEs.

From what I know it is somewhat similar in depth as Hale's book: i.e. covers some of the same topics Dan Blazevski listed below.

Answer 8

"Differential Equations: A Dynamical Systems Approach" by Hubbard and West (parts 1 and 2) are very pleasant reads for people with a fairly pure bent.

My initial exposure to differential equations was from an instructor that had taught so many service courses he appeared to be incapable of giving a conceptual overview of any subject. So my opinion of differential equations hit an early artificial low point. But Hubbard's books are very cheery in comparison.

Smale and Hirsch's "Differential Equations, Dynamical Systems and Linear Algebra" is quite a pleasant read. More dry than Hubbard and West, but that's not always bad.

I might have some other suggestions later...

I've often wondered if there were any good textbook accounts of the local-Lie-groupoid-of-symmetries approach to differential equations.

Answer 9

A classic is Coddington and Levinson. There is also a much simpler and smaller book by Coddington alone.

Answer 10

This is an old question, but it was just bumped to the top and I noticed that my favorite ODE book wasn't listed.

Anyway, I highly recommend Hurewicz's beautiful little book "Lectures on ordinary differential equations". It's extremely short, efficient, and easy to read, and it contains everything a non-analyst needs to know about ODE's. It would probably be hard to teach a course from it, but for self-study it is perfect.

Answer 11

It sounds like you also want an introduction to differential geometry, as well as a good grounding in ODE's. As an undergraduate, I had Martin Braun's book on differential equations and their applications, and Barrett O'Neill's Elementary Differential Geometry. They should be quite approachable, and a thorough reading should give you enough background for later courses.

I recommend doing some of the computations, because knowing some of the numerical analysis issues can be important, even though they are addressed less than superficially if at all in these books. Also, it's important that you "feel you could start doing calculus on a Moebius strip", at least locally, even if you don't actually do it. Such a feeling can give one comfort when one approaches the subject in depth. I missed having such a bedrock in some of my analysis/PDE courses; this may be why I ended up doing more algebra and logic instead.

Gerhard "Ask Me About System Design" Paseman, 2010.06.18

Answer 12

I suggest to give a look at the following notes

http://www.mat.univie.ac.at/~gerald/ftp/book-ode/

Answer 13

This isn't a direct answer to your question (I don't have a good book recommendation because that's not my field), but if there is a higher level course on differential equations or dynamics of some sort that interests you more, you might want to try petitioning to get that to count for your requirement instead. I did that both at my undergrad institution and grad institution and it was always approved. Generally if you're interested in taking a harder course (no need to mention to your advisor about the side benefit of it being less computational if you don't want) and you're not shooting yourself in the foot by doing so (i.e. you have the prereqs), they're unlikely to say no.