Differential Equations: A Modeling Perspective

I used a preliminary version of this book when I was a student, and was later involved for 2 months with doing edits for the first edition. I am now teaching a differential equations course out of a different textbook. I keep going back to this book for examples to use, or for better ways to present material. It's a very good book, and the models that I was exposed to while studying from this book have significantly affected the course of my research since then. A few words of warning: this book assumes that the students are comfortable with mathematical concepts and general science. If your students struggle with the fact that (a+b)^2 is not a^2 + b^2 or if they don't understand why e^{a+b} is not e^a + e^b, they are going to have a very hard time with this book. On the other hand, if you want your students to come out of the class knowing how to look at a physical problem and then figure out what the equations are that underlie it, and then to solve them, then this is a good book to use. Having read the reviews out there, some responses to the negative reviews are in order. The most common criticisms I've seen are either that there aren't enough theorems and proofs, or that there is too much guessing going on. This book, in its title, states that it is a modeling perspective - if you want a theorem-proof book you can definitely find them. However, if you want to teach physicists and engineers and applied mathematicians the skills they need, then spending all your time on theorems and proofs isn't the right approach -- a colleague of mine once told me he didn't learn how to use an integrating factor until graduate school, but he knew all sorts of things about whether a solution existed or was unique. If you are concerned that there is too much emphasis on "guessing", then you haven't read many ODE textbooks. Most textbooks I've seen suggest that you guess a solution, and then after the fact turn to the Wronskian or some similar argument to show that the guess will work. My experience with Borrelli and Coleman was that I constructed the general solution form, and then I understood why the solution takes a form and so I have no problem making that guess. To summarize: this is not an easy book intended for students who need to be constantly reminded of concepts from high school algebra. It assumes that the students have a moderate level of mathematical sophistication, and that they are comfortable with the idea that differential equations underlie most physical processes. If your intention is that the students understand how to find those equations and how to work with them in a wide variety of contexts, then this is the right book.

This elegantly written text broke down some barriers that I had while taking Differential Equations in college. Finally someone understands the way that I learn things. A modelling perspective is what I needed to understand these previously confusing concepts. My thanks to the authors!

First, I would like to warn students wishing to introduce themselves to Differential Equations (DEs) to avoid this book if they do not have a strong mathematical background. The book does not use a linear approach common in other books, and tends to integrate DEs with other subjects such as engineering, physics, biology, and other fields of mathematics while spreading out the topics. At times, the book tends to skip algebraic steps, making it confusing for those who are not as strong in algebra. First-timers will find this book hard to follow on their own, but with the guidance of a good professor, it should not be too hard to tackle. Unlike other textbooks that teaches one topic and proceeds to give thousands of examples, this book does not do that. Instead, it uses the "modeling approach" to show how DEs relates to other fields of science. A person interested in mathematics and science will find this book very intriguing because of how well it relates math to everything else in life. Problems in this book are generally harder than problems of average DEs books, but once the students become used to them, it will be much more helpful. Not only does this book help the students understand the new concept of DEs, it aids them in reviewing the calculus and algebra (along with complex numbers, linear algebra, and trigonometry). Overall, this is a perfect book for someone who likes math but finds most "advanced" high school textbooks too simple. This book does not merely repeat examples until the student understands the material. The student is expected to work out practice problems on his own and check his answers with the back of the book. This book uses proofs to discover theorems instead of just stating it while leaving the motivated student perplexed on how mathematicians came up with the theorem. Finally, this book includes the ODE (Ordinary Differential Equations) CD, which assists the student in the learning process with visual aids and graphic demonstrations. While this program may not be as powerful as Mathematica or Maple, it runs much faster and is more convenient in learning DEs.

This text teaches Differential Equations using the models and tools that both mathematicians and engineers use. It uses the modeling process to show how the DEs are developed. It uses numerous examples showing how to understand a DEs behavior using numerical solvers as well as standard analytical techniques and (when they exist) solution formulas.Because it doesn't use a cookbook approach to solving DEs, it isn't restricted to just those DEs for which there is a standard solution method. Since it can deal with this wider variety of DEs, it can introduce more interesting models (e.g., the autocatylator) which it would otherwise ignore. The graphs used to illustrate these models are easy to understand, incredibly detailed, and very pretty.This book gives a very complete introduction to Differential Equations and to the models that they describe.--A student at Harvey Mudd College

This text places the study of differential equations in the context of applications (modeling). It incorporates computing in thoughtful, relevant and helpful ways. Theoretical results are presented in a manner which is both meaningful in terms of usefullnes and appropriate to the level of sophistication of the anticipated college sophomore audience. Applications are drawn broadly from the natural sciences. The writing style is lively and direct. The level of sophistication is very appropriate to those who have completed a solid course in linear algebra, in contrast to the cookbook style of more traditional texts.