Ordinary Differential Equations (Dover Books on Mathematics)

First, some information about myself. I am a sophomore in college. I took a intro differential equations course last semester. I found it frustrating - the course covered many topics but none quite in depth as I would have liked. I am an engineer, and engineers are not supposed to "care" about the theory, only how to apply it, but I have a certain fascination with differential equations that was definitely not satisfied by the class I took. The textbook we used, Boyce and DiPrima, did not help matters. It was convoluted, spending whole pages trying to explain a concept, chock full of referrals to formulas a few pages back, interspersed with pretty pictures. In short, I appreciated what the authors tried to do, but it did not help me understand differential equations adequately. But alas, I digress. This is not a review of Boyce and DiPrima.Anyway, I began searching around for a book that would let me learn DE's the right way. This book came up in a recommendation, and I decided to try it after reading all the positive reviews about it. I think it does a fine job of living up to its reviews. The material is presented in a very clear, very accesible manner. The book is divided into lessons. Each lesson covers a specific topic. I am currently going through lesson 20, n-th order linear homogeneous ODE's with constant coefficients. The authors give a general overview and discuss briefly that e^mx is a solution to all of these equations provided they have constant coefficients. Then they give the three cases of concern - real distinct roots, real repeated roots, and complex roots. Each of these cases gets its own sublesson, starting off with a generalized equation, a proof, and an example. This isn't so different from what other textbooks do, but something about the uncluttered text, the effort that the authors put into explaining every nontrivial step of a proof, and the organization greatly appeals to me. As icing on the cake, at the end of every lesson is about 40 practice problems...with solutions to every one of them on the following page. Granted the solutions do not have steps, but the material is covered so throughly that a glance back is all you need to solve them.I'll give an example of how thorough the book is compared to Boyce & DiPrima using repeated roots cropping up in characterstic equations of second order homogenous ODE's. Say the root has value m and A and B are constants; the general solution to such an equation is y = Ae^(mx) + Bxe^(mx). In Boyce & DiPrima, the solution is presented in a stupid manner. The authors use an analogy to a first order equation to try and explain why xe^(ax) appears. The fact that I don't even remember the proof is testament to how poorly the topic was explained. In this book, the authors explain that y = Ae^(mx) + Be^(mx) is NOT a solution because the function Ae^(mx) is NOT independent of Be^(mx), and all solutions to n-th order linear homogenous ODE's REQUIRE a solution composed of a basis of n independent functions. Since e^(mx) cannot be used twice, there has to be another function besides e^(mx) that satisfies the differential equation y'' - 2my' + (m^2)y = 0 (of which m is a repeated root). They suggest y = u(x)e^(mx), and substitute this into the aforementioned differential equation. Then it is just a matter of finding u(x). It turns out that u''(x) = 0, so u(x) = B + Cx, Suddenly, it's all clear. The solution is thus y = Ae^(mx) + (B + Cx)e^(mx). But there's more. If the root is repeated 3 times, then the solution becomes y = Ae^(mx) + (B + Cx + Dx^2)e^(mx). And if it's repeated four times...etc. The authors make sure to cover every avenue of curiosity that one might have, in depth.Unlike Boyce & DiPrima, I'll remember that proof for a long time to come. I doubt many other convential DE textbooks present their topics with this much clarity and depth. And that was just one lesson. There are 65 lessons in the 800+ pages of this book. IMHO, the best way to take advantage of this book is to get a notebook, pencil, and paper, sit down at a table, pick a lesson, and go along with every derivation in your notebook. Then do every exercise and check the provided solutions. That's what I'm doing, anyway. It's what makes this book is ideal for self-learners. If you want pictures, go buy an overpriced college textbook. If you want substance and understanding, get this.

I took ODE this semester, and I was liking the subject until I got to read the textbooks assigned to it. It is impressive how the world is filled with giant text books that are absolutely dull and useless and extremely expensive. Luckly I have always been fond of Amazon, so I searched "Ordinary Differential Equations" and came upon this book, which at first glance looks tiny and unpromising, but trust me, this little beast doesn't only talk about ODE, it takes the subject, makes it its own, and in the most elegant of fashions transmits the knowledge so well that it even if I live in Ecuador and English is only my second language, I could grasp all what was necessary to, not only pass ODE, but to take my knowledge and apply it to computer programming right away.Trust me, if a book teaches so well that you can go ahead and apply it just like that, it is something special.Now strictly speaking on it's qualities:First, the book is a breeze to read, you will not find yourself reading back again through the text because of the lack of good pedagogy, but be aware, the writer does not bother to make you laugh either (a quality most serious books should not have, but I like what Stephen Prata did on C++ Primer Plus). Secondly, Ordinary Differential Equations has all that you will probably need for the subject. Check the MIT Open Course Ware, I downloaded the exams on the web page and did them singlehandedly, only with what this book taught me. Actually, you'll see lots of other topics that MIT doesn't even cover, for example it has a very interesting section on numerical methods.Something that has to be mentioned is that this book covers a great amount of material in a excellent order and pace. The writer never assumes that you are a genius on calculus, so he always makes sure to guide you, holding your hand on each topic, repeating theorems already mentioned to refresh your head, not skipping too many steps when solving examples. This feature is seen at it's best in the Series Methods section of the book. Also, the amount of problems is wonderful, they all have solutions and are right next to the problems, unlike the convention, which gives solutions only to the odd number problems and has them written at the very end of the book, something that I hate, for the constant page turning greatly damages the book. Don't you worry, the writer solves many examples and each subject, explaining everything so you can work on the problem set rather easily.The only setbacks that I noticed on this book are that, when teaching the prerequisites to a subject, it doesn't bother to demonstrate the theorems (which is fine by me, because you should already know that stuff in the fist place), and it doesn't have all the fancy graphics that the outrageously expensive ODE books have (for this I use Matlab or Mathematica, so I also don't care about his). You also have to consider that his books is quite old, and the numerical methods are a bit dated, still, any good teacher will fill you in with the little updates made to the subject.All in all this book is nothing short of amazing, I give it all my fingers up to anyone who is taking ODE or wants an awesome reference book. I found it easy to read, precise, and vast. This book will probably do you more justice than anything worth >$100.

I've become a fan over the Dover books as a quick pickup for math. They are inexpensive, usually pretty solid technically, and understand that you are probably not a mathematician and need your hand held long enough to find your sea legs. You're busy. You need to know this stuff so you can do something else. You're an engineer of some kind with tight deadlines. You're wearing a black belt with brown shoes. You drink way too much coffee and really need to let your kids know you love them more than you do.I know your kind. And it'll be okay.That aside, make sure you have a solid footing on your calculus, because if you didn't do well with calculus, you'll suck with this too. The mistakes just multiply. Again, the authors understand you're not a mathematician, but they know you're not an idiot either and don't need your hand held. If you've got those basic prerequisites (1. Calc 2. Not an idiot) and need to get in the know on DiffEQ, this is a good one.

After finishing James Stewart Calculus 9th Ed. I find this book great to study before moving on to Kreyszig's Advanced Engineering Mathematics. So far I've read the first 110 pages of this book and I think the examples and explanations are quite easy to follow compared to other books on ODEs. The book gives Full answers of the exercises which is helpful.