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The Theory of Differential Equations: Classical and Qualitative

Walter G. Kelley and Allan C. Peterson
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Florin Catrina
, on

This is a very good book on Differential Equations. It is the kind of book I would use in the classroom as well as recommend to a student for independent study. I can see it used as textbook for a course in Differential Equations, year-long if Calculus and Linear Algebra background needs to be reinforced, or in a one semester course for math majors.

The book does not include series solutions, the Laplace transform, or numerical methods, so it may not be immediately useful to engineering majors. On the other hand, it builds the theory of Differential Equations, and it does it well. Floquet theory is in Chapter 2, autonomous systems are discussed in Chapter 3, and Chapter 4 contains perturbation methods. It may seem that these topics are out of place in a textbook for a first class in Differential Equations, but through their choice of examples and exercises the authors make the topics flow at a natural pace.

There are nice small side topics, such as the construction of sine and cosine functions and the proof of the basic trigonometric identities based solely on the existence and uniqueness theorem, as well as less usual topics, such as factorization theorems.

Chapters 5 to 8 take the reader through Sturm-Liouville problems, the calculus of variations, higher order systems, nonlinear ODE’s, and classical existence and uniqueness theorems. These topics are sufficiently rich for each one of them to be the subject of a separate book. The present text deals with them to the point where substantial theorems can be discussed, and at the same time leaves the reader wanting to find out what more can be said. The references are appropriate and point to specific pages in the classic texts (Coddington and Levinson, Hartman, etc.).

I believe that instructors would enjoy teaching from this book, and that students would be able to study from it (either through a class or independently) at a good pace. And they would learn a lot about differential equations.

Florin Catrina is Assistant Professor of Mathematics at St. John's University in Queens, New York.


Chapter 1 First-Order Differential Equations
1.1 Basic Results
1.2 First-Order Linear Equations
1.3 Autonomous Equations
1.4 Generalized Logistic Equation
1.5 Bifurcation
1.6 Exercises

Chapter 2 Linear Systems
2.1 Introduction
2.2 The Vector Equation x' = A(t)x
2.3 The Matrix Exponential Function
2.4 Induced Matrix Norm
2.5 Floquet Theory
2.6 Exercises

Chapter 3 Autonomous Systems
3.1 Introduction
3.2 Phase Plane Diagrams
3.3 Phase Plane Diagrams for Linear Systems
3.4 Stability of Nonlinear Systems
3.5 Linearization of Nonlinear Systems
3.6 Existence and Nonexistence of Periodic Solutions
3.7 Three-Dimensional Systems
3.8 Differential Equations and Mathematica
3.9 Exercises

Chapter 4 Perturbation Methods
4.1 Introduction
4.2 Periodic Solutions
4.3 Singular Perturbations
4.4 Exercises

Chapter 5 The Self-Adjoint Second-Order Differential Equation
5.1 Basic Definitions
5.2 An Interesting Example
5.3 Cauchy Function and Variation of Constants Formula
5.4 Sturm-Liouville Problems
5.5 Zeros of Solutions and Disconjugacy
5.6 Factorizations and Recessive and Dominant Solutions
5.7 The Riccati Equation
5.8 Calculus of Variations
5.9 Green’s Functions
5.10 Exercises

Chapter 6 Linear Differential Equations of Order n
6.1 Basic Results
6.2 Variation of Constants Formula
6.3 Green’s Functions
6.4 Factorizations and Principal Solutions
6.5 Adjoint Equation
6.6 Exercises

Chapter 7 BVPs for Nonlinear Second-Order DEs
7.1 Contraction Mapping Theorem (CMT)
7.2 Application of the CMT to a Forced Equation
7.3 Applications of the CMT to BVPs
7.4 Lower and Upper Solutions
7.5 Nagumo Condition
7.6 Exercises

Chapter 8 Existence and Uniqueness Theorems
8.1 Basic Results
8.2 Lipschitz Condition and Picard-Lindelof Theorem
8.3 Equicontinuity and the Ascoli-Arzela Theorem
8.4 Cauchy-Peano Theorem
8.5 Extendability of Solutions
8.6 Basic Convergence Theorem
8.7 Continuity of Solutions with Respect to ICs
8.8 Kneser’s Theorem
8.9 Differentiating Solutions with Respect to ICs
8.10 Maximum and Minimum Solutions
8.11 Exercises

Solutions to Selected Problems