You are here

Fundamentals of Differential Equations

R. Kent Nagle, Edward B. Saff, and Arthur David Snider
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
, on

This is the short version of the classic textbook, having only ten chapters, in contrast to its longer brother, whose title continues with the words “and boundary value problems.” As this is the eighth edition of the book, the reviewer’s task is twofold. He needs to write something for readers who have never seen the book under review, but also something for readers who want to know what is new in the most recent edition.

For the former, this is a solid introductory textbook into differential equations. Even in this short version, there is more than enough for a one-semester course. In fact, such a course can be taught using only chapters 1, 2, 4, 6, 7, and 8, which cover basic notions, linear, exact and separable equations, second order equations and their generalizations to higher order, Laplace transforms, and power series. Chapters 3 and 5 contain applications of the theory learned in the other chapters. The topical coverage is mathematically rigorous yet down-to-earth. There are plenty of exercises, half of which have their answers in the book. In case you have time to go further, which is not likely, the last two chapters contain matrix methods and partial differential equations.

As far as the what is new in the most recent edition, not that much, especially in the core parts that many instructors will use. There are some additions in other parts of the book, such as a review of integration in the appendix, several new projects, and extra material on eigenfunctions and eigenvalues. The latter will not be used by instructors whose students are to take linear algebra after differential equations.

To summarize, if you liked the previous editions, you will like this one, and if you have not seen any earlier editions, you might as well check this one out. If you do decide to use the book, make sure your students know whether you use the longer or shorter version — some will buy the wrong one every year.

Miklós Bóna is Professor of Mathematics at the University of Florida.

1. Introduction

1.1 Background

1.2 Solutions and Initial Value Problems

1.3 Direction Fields

1.4 The Approximation Method of Euler

           Chapter Summary

           Technical Writing Exercises

           Group Projects for Chapter 1

           A. Taylor Series Method

           B. Picard's Method

           C. The Phase Line

2. First-Order Differential Equations

2.1 Introduction: Motion of a Falling Body

2.2 Separable Equations

2.3 Linear Equations

2.4 Exact Equations

2.5 Special Integrating Factors

2.6 Substitutions and Transformations

           Chapter Summary

           Review Problems

           Technical Writing Exercises

           Group Projects for Chapter 2

           A. Oil Spill in a Canal

           B. Differential Equations in Clinical Medicine

           C. Torricelli's Law of Fluid Flow

           D. The Snowplow Problem

           E. Two Snowplows

           F. Clairaut Equations and Singular Solutions

           G. Multiple Solutions of a First-Order Initial Value Problem

           H. Utility Functions and Risk Aversion

           I. Designing a Solar Collector

           J. Asymptotic Behavior of Solutions to Linear Equations

3. Mathematical Models and Numerical Methods Involving First Order Equations

3.1 Mathematical Modeling

3.2 Compartmental Analysis

3.3 Heating and Cooling of Buildings

3.4 Newtonian Mechanics

3.5 Electrical Circuits

3.6 Improved Euler's Method

3.7 Higher-Order Numerical Methods: Taylor and Runge-Kutta

           Group Projects for Chapter 3

           A. Dynamics of HIV Infection

           B. Aquaculture

           C. Curve of Pursuit

           D. Aircraft Guidance in a Crosswind

           E. Feedback and the Op Amp

           F. Bang-Bang Controls

           G. Market Equilibrium: Stability and Time Paths

           H. Stability of Numerical Methods

           I. Period Doubling and Chaos

4. Linear Second-Order Equations

4.1 Introduction: The Mass-Spring Oscillator

4.2 Homogeneous Linear Equations: The General Solution

4.3 Auxiliary Equations with Complex Roots

4.4 Nonhomogeneous Equations: The Method of Undetermined Coefficients

4.5 The Superposition Principle and Undetermined Coefficients Revisited

4.6 Variation of Parameters

4.7 Variable-Coefficient Equations

4.8 Qualitative Considerations for Variable-Coefficient and Nonlinear Equations

4.9 A Closer Look at Free Mechanical Vibrations

4.10 A Closer Look at Forced Mechanical Vibrations

           Chapter Summary

           Review Problems

           Technical Writing Exercises

           Group Projects for Chapter 4

           A. Nonlinear Equations Solvable by First-Order Techniques

           B. Apollo Reentry

           C. Simple Pendulum

           D. Linearization of Nonlinear Problems

           E. Convolution Method

           F. Undetermined Coefficients Using Complex Arithmetic

           G. Asymptotic Behavior of Solutions

5. Introduction to Systems and Phase Plane Analysis

5.1 Interconnected Fluid Tanks

5.2 Elimination Method for Systems with Constant Coefficients

5.3 Solving Systems and Higher-Order Equations Numerically

5.4 Introduction to the Phase Plane

5.5 Applications to Biomathematics: Epidemic and Tumor Growth Models

5.6 Coupled Mass-Spring Systems

5.7 Electrical Systems

5.8 Dynamical Systems, Poincaré Maps, and Chaos

           Chapter Summary

           Review Problems

           Group Projects for Chapter 5

           A. Designing a Landing System for Interplanetary Travel

           B. Spread of Staph Infections in Hospitals-Part 1

           C. Things That Bob

           D. Hamiltonian Systems

           E. Cleaning Up the Great Lakes

6. Theory of Higher-Order Linear Differential Equations

6.1 Basic Theory of Linear Differential Equations

6.2 Homogeneous Linear Equations with Constant Coefficients

6.3 Undetermined Coefficients and the Annihilator Method

6.4 Method of Variation of Parameters

           Chapter Summary

           Review Problems

           Technical Writing Exercises

           Group Projects for Chapter 6

           A. Computer Algebra Systems and Exponential Shift

           B. Justifying the Method of Undetermined Coefficients

           C. Transverse Vibrations of a Beam

7. Laplace Transforms

7.1 Introduction: A Mixing Problem

7.2 Definition of the Laplace Transform

7.3 Properties of the Laplace Transform

7.4 Inverse Laplace Transform

7.5 Solving Initial Value Problems

7.6 Transforms of Discontinuous and Periodic Functions

7.7 Convolution

7.8 Impulses and the Dirac Delta Function

7.9 Solving Linear Systems with Laplace Transforms

           Chapter Summary

           Review Problems

           Technical Writing Exercises

           Group Projects for Chapter 7

           A. Duhamel's Formulas

           B. Frequency Response Modeling

           C. Determining System Parameters

8. Series Solutions of Differential Equations

8.1 Introduction: The Taylor Polynomial Approximation

8.2 Power Series and Analytic Functions

8.3 Power Series Solutions to Linear Differential Equations

8.4 Equations with Analytic Coefficients

8.5 Cauchy-Euler (Equidimensional) Equations

8.6 Method of Frobenius

8.7 Finding a Second Linearly Independent Solution

8.8 Special Functions

           Chapter Summary

           Review Problems

           Technical Writing Exercises

           Group Projects for Chapter 8

           A. Alphabetization Algorithms

           B. Spherically Symmetric Solutions to Shrödinger's Equation for the Hydrogen Atom

           C. Airy's Equation

           D. Buckling of a Tower

           E. Aging Spring and Bessel Functions

9. Matrix Methods for Linear Systems

9.1 Introduction

9.2 Review 1: Linear Algebraic Equations

9.3 Review 2: Matrices and Vectors

9.4 Linear Systems in Normal Form

9.5 Homogeneous Linear Systems with Constant Coefficients

9.6 Complex Eigenvalues

9.7 Nonhomogeneous Linear Systems

9.8 The Matrix Exponential Function

           Chapter Summary

           Review Problems

           Technical Writing Exercises

           Group Projects for Chapter 9

           A. Uncoupling Normal Systems

           B. Matrix Laplace Transform Method

           C. Undamped Second-Order Systems

           D. Undetermined Coefficients for System Forced by Homogeneous

10. Partial Differential Equations

10.1 Introduction: A Model for Heat Flow

10.2 Method of Separation of Variables

10.3 Fourier Series

10.4 Fourier Cosine and Sine Series

10.5 The Heat Equation

10.6 The Wave Equation

10.7 Laplace's Equation

           Chapter Summary

           Technical Writing Exercises

           Group Projects for Chapter 10

           A. Steady-State Temperature Distribution in a Circular Cylinder

           B. A Laplace Transform Solution of the Wave Equation

           C. Green's Function

           D. Numerical Method for u=f on a Rectangle


A. Newton's Method

B. Simpson's Rule

C. Cramer's Rule

D. Method of Least Squares

E. Runge-Kutta Procedure for n Equations

Answers to Odd-Numbered Problems