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

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

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I am having exactly this issue currently. I live in Australia, and as a result there are not that many mathematics courses offered at my university (in particular, there are no graduate courses). In order to graduate i have had to take many grunt courses –  Daniel Barter Jun 19 '10 at 7:29
WHY you're torturing yourself with Harsthorne when Shafarevich's 2 volume classic and so many good lecture notes-such as Ravi Vakil's terrific notes at Stanford-are available.Email Dr.Vakil and nicely ask for a copy of the current version of the notes and I'm sure he'll send you a copy.You'll find them VERY helpful. –  Andrew L Jun 19 '10 at 7:32
I would echo the answers suggesting Arnol'd, particularly the DE book. I never took an ODE course in my life and when I wanted a reference this was the first one I bought. –  Steve Huntsman Jun 19 '10 at 12:56
Community wiki? –  Grétar Amazeen Jun 23 '10 at 2:48
A brief follow-up to Andrew L's kind comments: many people write asking for my differential equations notes, but these notes are at the (serious) PhD level in algebraic geometry. (And for the algebraic geometry notes, you can get them from math216.wordpress.com.) –  Ravi Vakil Jan 1 at 17:18

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.

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I apologize if I sounded condescending in assessing the course I am taking, that wasn't my intention. I wanted merely to convey that it is intended for engineering-types who openly disparage pure maths. I am quite sure that the mathematically substantive material will be largely rolled over so that the computational requirements of the students occupying the course can be met. Yours seems like a good list, and I do appreciate it. I must ask, however (given your statement above) whether you find Spivak's 'Calculus On Manifolds' which I referenced before to not have the proper context. –  lambdafunctor Jun 19 '10 at 8:00
Also, Mr. Protsak, I do completely agree with you about the disparity between those who actually 'do' the mathematics and those who fixate themselves on abstractions and higher constructions. I try to walk that fine line, remaining 'inspired' by the exciting research and such which goes on in the mathematical community proper while getting my hands dirty inculcating myself (often force-feeding) the essential foundational material with which I may improve my mathematical maturity, despite how 'computational' or dry it may seem. I know that I must walk before I can sprint. –  lambdafunctor Jun 19 '10 at 8:07
@Victor Amen,brother. You and I are very much on the same wavelength here.I really can't improve on your response.So I won't try. –  Andrew L Jun 19 '10 at 8:27
You don't need to apologize to me, but it may be a good idea to tone down your post a little and emphasize the constructive side. Spivak is a good book for learning calculus on manifolds (mostly, integral calculus as I recall) for its own sake, but your question was about differential equations, right? Arnold's "Mathematical methods" really shows you where it comes from and where it leads (it's been a while since I opened it, but that's my recollection). It's a bit advanced (that's why I put it last on the list); if you liked Kostrikin and Manin, I hope you'll like Arnold's ODE book (#3). –  Victor Protsak Jun 19 '10 at 8:38
Piskunov, my favorite since high school. –  Unknown Jun 19 '10 at 19:07
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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.

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I could not agree more with these choices! –  José Figueroa-O'Farrill Jun 23 '10 at 2:39
I believe the union of classic Hirsh/Smale and it's second edition with Devaney may be the best intermediate ODE text that currently exists. –  Andrew L Jul 4 '10 at 2:32

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

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Dick,I've read parts of the book online.It's not as comprehensive as some of the intermediate to advanced texts-like Arnold or Hirsh/Smale-but it is very readable and more user friendly. It's much better suited for an honors undergraduate course on ODE's then these books and thanks for writing a book at this level. I strongly suspect finding the existing choices for such a course either too easy or too hard lead to the writing of this one. I'd love to try and teach such a course one day out of a later edition. (Hopefully there will be several!) –  Andrew L Jul 4 '10 at 2:30
Thank you so much for your suggestion, Mr. Palais; I have now had an opportunity to look over a lot of the material in your text and it seems very suitable to what I had in mind. I am sure that it will be a great primer for my course and hopefully improve my intuitions about the subject for future studies. –  lambdafunctor Jul 12 '10 at 7:09

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 then 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.

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ARRRRGHHHHH!!! PLEASE someone tell me why the punctuation and formatting never comes out on any of my posts and they all end up looking like a letter from the IRS!!! –  Andrew L Jun 19 '10 at 7:28
For a start, I suggest using the ENTER key to create shorter paragraphs. When the IRS uses short paragraphs, their letters seem more readable to me. Gerhard "Ask Me About System Design" Paseman, 2010.06.19 –  Gerhard Paseman Jun 19 '10 at 7:46
Thank you, Andrew; your comment and detailed list certainly gives me a lot to consider. I love that there exists some consensus here as to which few books to look for! I will have to go with the wisdom of the MO community here :) –  lambdafunctor Jun 19 '10 at 8:01
Andrew - 1) When separating into paragraphs you need to use TWO new lines rather than one 2) Always look at the preview before posting your answer, then fix it accordingly. –  danseetea Jun 19 '10 at 11:10
@Andrew L, I suggest you follow danseetea's advice --- look very carefully at the preview box before you post. Make sure that the paragraphs are separated properly, and that you've correctly put a space after each punctuation mark (commas and full stops). If there really is a discrepancy between what you type and what the preview shows or what finally displays, as you've claimed to me in email, please take a screen shot and send it to me. –  Scott Morrison Jun 19 '10 at 17:06

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.

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Yeah, Hurewicz's little book is a real treasure! –  drbobmeister Mar 31 '11 at 6:20

"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.

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Hmm, I recall hearing before that Smale and Hirsch was very good, so I might have to check it out. I'll look up Hubb. and West at the library, as well. Basically, I'm looking for a differential equations text which I will adore at least half as much for the subject as I do Kostrikin/Manin for linear algebra or Spivak for manifolds. I don't know how familiar you are with the structure of these particular texts (likely more than I am, because they are a canonical part of the literature), but they follow a "basic structures-->rigorous 'abstractification'-->glimpse at advanced topics"approach. –  lambdafunctor Jun 19 '10 at 7:22
@Ryan:For the "local-Lie-groupoid-of-symmetries approach" try 1) Elementary Lie group analysis and ordinary differential equations by N.H. Ibragimov(books.google.com/books?id=EWgZAQAAIAAJ), and 2) Applications of Lie Groups to Differential Equations by P.J. Olver (books.google.com/books?id=ACzC8sHg3jEC), 3) Differential equations: their solution using symmetries by H. Stephani (books.google.com/books?id=nFSJn7dIYysC). #1 is in fact an ODE textbook with symmetries intervowen into the story, #2 and #3 are more advanced and require some preliminary knowledge of ODEs and PDEs. –  mathphysicist Jun 19 '10 at 11:15

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

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Yes, the Coddington is great for future pure mathematicians who are interested in branches of mathematics OTHER than differential equations. –  Gerald Edgar Dec 14 '12 at 14:14

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.

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I'm taking graduate numerical analysis to fill space (it is truly just filling space, as I am interested in arithmetic/algebraic geometry and geometric representation theory, so it won't help me much in the future), so I feel comfortable with the computational aspect. I am looking for a more differential-geometric perspective on the subject, so your suggestions are very helpful. –  lambdafunctor Jun 19 '10 at 6:03

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.

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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.

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Hartman's book is quite difficult,Willie-it would be a very special undergraduate indeed who could read it and comprehend most of it.I think Arnold or Perko would be much better choices for strong undergraduates to begin with. –  Andrew L Jul 22 '10 at 2:07
The OP wanted a supplementary text for the theory and stuff, no? If the OP were asking for a book to teach an intro undergrad class from, I'd agree with you. But the OP's description of himself surely puts him in your category of "very special undergraduate". I do not disagree that it is hard going, but since an undergraduate course is unlikely to cover beyond the first 4 or 5 chapters of the book anyway, I don't think slow reading is a problem. –  Willie Wong Jul 22 '10 at 9:47

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.

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Right; I did that with graduate algebra, complex analysis, algebraic geometry (yet to take, but I have the go-ahead for this fall), etc. I figured that diff. eq. might be a nice reprieve from an otherwise dreadfully rigorous course load, but I want to actually learn something with pith along the way. I'm still weighing all of my options :) –  lambdafunctor Jun 19 '10 at 6:04

I suggest to give a look at the following notes

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

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