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Applied Differential Equations

Vladimir A. Dobrushkin
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Textbooks in Mathematics
[Reviewed by
William J. Satzer
, on

It is very hard to write a fresh introductory textbook on differential equations. There are dozens of similar books available now, many of them hard to distinguish from one another. The current book is a bit different. While it is not truly original, it gives the impression that the author went back to first principles and asked himself what a good introductory book on differential equations should have. What he produced is not the least bit trendy, but it has a sense of purpose and direction not always evident in comparable texts. (The author notes that the book would not have been written had students not “complained so much about other texts unleashed on them”.)

Two notable aspects of the book are its comprehensiveness and tight integration with applications. The author has a clear vision of his goals which include: showing that differential equations are essential for modeling real-life phenomena; promoting qualitative analysis as an essential tool for understanding differential equations; giving students a comprehensive and integrated picture of differential equations; introducing software packages commonly used for the solution and analysis of differential equations; and providing enough flexibility to allow instructors to design a curriculum that meets their individual goals.

The only prerequisite is a basic calculus course. The book is intended for sophomore or junior students in the sciences, engineering, economics and related fields. There is enough material here for two semesters — an introductory course followed by another with more advanced topics. But the book offers enough flexibility that intermediate level courses could also easily be accommodated.

The table of contents has no surprises. The topics and their order of treatment is not very different from scores of other books. Some of what distinguishes this from comparable texts is immediately evident (a lot more detail and many more exercises), but the differences are often more subtle (a better sense of showing the subject as an integrated whole, for example). True to his overall theme, the author starts right off with motivating applications. Then he immediately dives into a discussion of what’s meant by a solution, direction fields, existence and uniqueness of solutions, and a proof of Picard’s theorem. At the same time he begins introducing the software tools he will use throughout (MATLAB, Mathematica, Maple and Maxima).

One of the things about the book that did surprise me was the amount of space devoted to explicit or implicit solutions of first order equations (separable or exact equations, those amenable to solution with an integrating factor, ordinary linear systems, and then special equations like the Bernoulli or Riccati equations). This section is accompanied a large set of exercises to identify and solve equations like this.

Other sections of the book follow a similar pattern: applications, some theory, worked-out examples and application of software tools as appropriate. A section on numerical methods is mostly standard. The same is true of second and higher order linear differential equations, except that the Bessel equations get more attention than usual. Enough linear algebra is introduced to be able to handle systems of linear equations conveniently. But there are a few bonuses: a more detailed treatment of matrix functions and discussions of resolvent and spectral decomposition methods.

The book also includes a good treatment of the Laplace transform and an introduction to orthogonal expansions that is aimed particularly at Sturm-Liouville problems. The last chapter is a brief introduction to partial differential equations — mostly, the heat, wave and Laplace equations. It is clear that the author would have included more, but the book was already bursting at the seams.

There is much to like about this book — lucid writing, clear development of the basic ideas, and a very large number of exercises with a good range of difficulty. Weaknesses are relatively few. The narrative flow is sometimes disrupted by the inclusion of software code in line. While this can sometimes be desirable to link the development directly with the code, it is more often distracting. Nevertheless, in most aspects this is a very attractive text.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

First-Order Equations
Separable Equations
Equations with Homogeneous Coefficients
Exact Differential Equations
Integrating Factors
First-Order Linear Differential Equations
Equations Reducible to first Order
Existence and Uniqueness
Review Questions for Chapter 1

Applications of First Order ODE
Applications in Mathematics
Curves of Pursuit
Chemical Reactions
Population Models
Applications in Physics
Flow Problems
Review Questions for Chapter 2

Mathematical Modeling and Numerical Methods
Mathematical Modeling
Compartment Analysis
Difference Equations
Euler’s Methods
Error Estimates
The Runge-Kutta Methods
Multistep Methods
Error Analysis and Stability
Review Questions for Chapter 3

Second-order Equations
Second and Higher Order Linear Equations
Linear Independence and Wronskians
The Fundamental Set of Solutions
Equations with Constant Coefficients
Complex Roots
Repeated Roots. Reduction of Order
Nonhomogenous Equations
Variation of Parameters
Operator Method
Review Questions for Chapter 4

Laplace Transforms
The Laplace Transform
Properties of the Laplace Transform
Discontinuous and Impulse Functions
The Inverse Laplace Transform
Applications to Homogenous Equations
Applications to Non-homogenous Equations
Internal Equations
Review Questions for Chapter 5

Series of Solutions
Review of Power Series
The Recurrence
Power Solutions about an Ordinary Point
Euler Equations
Series Solutions Near a Regular Singular Point
Equations of Hypergeometric Type
Bessel’s Equations
Legendre’s Equation
Orthogonal Polynomials
Review Questions for Chapter 6

Applications of Higher Order Differential Equations
Boundary Value Problems
Some Numerical Methods
Dynamics of Rotational Motion
Harmonic Motion
Modeling: Forced Oscillations
Modeling of Electric Circuits
Some Variational Problems
Review Questions for Chapter 7

Appendix: Software Packages
Answers to Problems