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4.5
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.
