The Theory of Differential Equations: Classical and Qualitative (Paperback, 2, 2010)

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