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The Theory of Differential Equations: Classical and Qualitative

Table of Contents

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

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