The Theory of Differential Equations

For over 300 years, differential equations have served as an essential tool for describing and analyzing problems in many scientific disciplines. This carefully-written textbook provides an introduction to many of the important topics associated with ordinary differential equations. Unlike most textbooks on the subject, this text includes nonstandard topics such as perturbation methods and differential equations and Mathematica. In addition to the nonstandard topics, this text also contains contemporary material in the area as well as its classical topics. This second edition is updated to be compatible with Mathematica, version 7.0. It also provides 81 additional exercises, a new section in Chapter 1 on the generalized logistic equation, an additional theorem in Chapter 2 concerning fundamental matrices, and many more other enhancements to the first edition. This book can be used either for a second course in ordinary differential equations or as an introductory course for well-prepared students. The prerequisites for this book are three semesters of calculus and a course in linear algebra, although the needed concepts from linear algebra are introduced along with examples in the book. An undergraduate course in analysis is needed for the more theoretical subjects covered in the final two chapters.

ISBN 1441957820   Pages 434 ISBN13 9781441957825 (What's this?)   Weight (grammes) 608 Publisher Springer-Verlag New York Inc.   Published in New York, NY Imprint Springer-Verlag New York Inc.   Series title Universitext Format Paperback   Height (mm) 234 Publication date 27 Apr 2010   Width (mm) 156 Library of Congress 2010924820   Spine width (mm) 22 DEWEY 515.35   Academic level Professional / Scholarly DEWEY edition DC22

Preface.- 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.- Bibliography.- Index

From the reviews of the second edition: "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 ! or in a one semester course for math majors. ! instructors would enjoy teaching from this book, and that students would be able to study from it ! at a good pace. And they would learn a lot about differential equations." (Florin Catrina, The Mathematical Association of America, November, 2010)

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