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 a chapter on perturbation methods and a section in Chapter 3 that shows how to solve differential equations using Mathematica codes. 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, and all Mathematica codes are in the book itself. This new edition 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 further 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.

Inhalt

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 ConvergenceTheorem.- 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