A Modern Introduction to Differential Equations

Chapter 1 Introduction to Differential EquationsIntroduction1.1 Basic Terminology1.2 Solutions of Differential Equations1.3 Initial-Value Problems and Boundary-Value ProblemsSummaryProject 1-1 Chapter 2 First-Order Differential Equations Introduction2.1 Separable Equations 2.2 Linear Equations2.3 Compartment Problems2.4 Slope Fields 2.5 Phase Lines and Phase Portraits 2.6 Equilibrium Points: Sinks, Sources, and Nodes 2.7 Bifurcations 2.8 Existence and Uniqueness of SolutionsSummaryProject 2-1Project 2-2Chapter 3 The Numerical Approximation of Solutions Introduction3.1 Euler’s Method3.2 The Improved Euler Method 3.3 More Sophisticated Numerical Methods: Runge-Kutta and OthersSummary Project 3-1 Chapter 4 Second- and Higher-Order EquationsIntroduction4.1 Homogeneous Second-Order Linear Equations with Constant Coefficients4.2 Nonhomogeneous Second-Order Linear Equations with Constant Coefficients4.3 The Method of Undetermined Coefficients4.4 Variation of Parameters 4.5 Higher-Order Linear Equations with Constant Coefficients4.6 Higher-Order Equations and Their Equivalent Systems4.7 The Qualitative Analysis of Autonomous Systems4.8 Spring-Mass Problems4.9 Existence and Uniqueness4.10 Numerical Solutions SummaryProject 4-1 Chapter 5 Systems of Linear Differential Equations Introduction 5.1 Systems and Matrices5.2 Two-Dimensional Systems of First-Order Linear Equations5.3 The Stability of Homogeneous Linear Systems: Unequal Real Eigenvalues5.4 The Stability of Homogeneous Linear Systems: Equal Real Eigenvalues5.5 The Stability of Homogeneous Linear Systems: Complex Eigenvalues5.6 Nonhomogeneous Systems5.7 Generalizations: The n × n Case (n ≥ 3) Summary Project 5-1 Project 5-2Chapter 6 The Laplace TransformIntroduction 6.1 The Laplace Transform of Some Important Functions6.2 The Inverse Transform and the Convolution 6.3 Transforms of Discontinuous Functions6.4 Transforms of Impulse Functions—The Dirac Delta Function6.5 Transforms of Systems of Linear Differential Equations6.6 A Qualitative Analysis via the Laplace TransformSummary Project 6-1Chapter 7 Systems of Nonlinear Differential EquationsIntroduction7.1 Equilibria of Nonlinear Systems7.2 Linear Approximation at Equilibrium Points7.3 The Poincaré-Lyapunov Theorem 7.4 Two Important Examples of Nonlinear Equations and Systems7.5 Van Der Pol’s Equation and Limit CyclesSummaryProject 7-1Appendix A Some Calculus Concepts and Results A.1 Local Linearity: The Tangent Line Approximation A.2 The Chain Rule A.3 The Taylor Polynomial / Taylor Series A.4 The Fundamental Theorem of Calculus (FTC) A.5 Partial Fractions A.6 Improper Integrals A.7 Functions of Several Variables / Partial Derivatives A.8 The Tangent Plane: The Taylor Expansion of F(x, y) Appendix B Vectors and Matrices B.1 Vectors and Vector Algebra; Polar Coordinates B.2 Matrices and Basic Matrix Algebra B.3 Linear Transformations and Matrix Multiplication B.4 Eigenvalues and Eigenvectors Appendix C Complex Numbers C.1 Complex Numbers: The Algebraic ViewC.2 Complex Numbers: The Geometric ViewC.3 The Quadratic Formula C.4 Euler’s FormulaAppendix D Series Solutions of Differential Equations D.1 Power Series Solutions of First-Order Equations D.2 Series Solutions of Second-Order Linear Equations: Ordinary Points D.3 Regular Singular Points: The Method of Frobenius D.4 The Point at Infinity D.5 Some Additional Special Differential EquationsAnswers/Hints to Odd-Numbered ExercisesIndex