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• Title Page
• Copyright Page
• Acknowledgments
• CONTENTS
• Application Modules
• Preface
• CHAPTER 1 First-Order Differential Equations
• 1.1 Differential Equations and Mathematical Models
• 1.2 Integrals as General and Particular Solutions
• 1.3 Slope Fields and Solution Curves
• 1.4 Separable Equations and Applications
• 1.5 Linear First-Order Equations
• 1.6 Substitution Methods and Exact Equations
• CHAPTER 2 Mathematical Models and Numerical Methods
• 2.1 Population Models
• 2.2 Equilibrium Solutions and Stability
• 2.3 Acceleration–Velocity Models
• 2.4 Numerical Approximation: Euler’s Method
• 2.5 A Closer Look at the Euler Method
• 2.6 The Runge–Kutta Method
• CHAPTER 3 Linear Equations of Higher Order
• 3.1 Introduction: Second-Order Linear Equations
• 3.2 General Solutions of Linear Equations
• 3.3 Homogeneous Equations with Constant Coefficients
• 3.4 Mechanical Vibrations
• 3.5 Nonhomogeneous Equations and Undetermined Coefficients
• 3.6 Forced Oscillations and Resonance
• 3.7 Electrical Circuits
• 3.8 Endpoint Problems and Eigenvalues
• CHAPTER 4 Introduction to Systems of Differential Equations
• 4.1 First-Order Systems and Applications
• 4.2 The Method of Elimination
• 4.3 Numerical Methods for Systems
• CHAPTER 5 Linear Systems of Differential Equations
• 5.1 Matrices and Linear Systems
• 5.2 The Eigenvalue Method for Homogeneous Systems
• 5.3 A Gallery of Solution Curves of Linear Systems
• 5.4 Second-Order Systems and Mechanical Applications
• 5.5 Multiple Eigenvalue Solutions
• 5.6 Matrix Exponentials and Linear Systems
• 5.7 Nonhomogeneous Linear Systems
• CHAPTER 6 Nonlinear Systems and Phenomena
• 6.1 Stability and the Phase Plane
• 6.2 Linear and Almost Linear Systems
• 6.3 Ecological Models: Predators and Competitors
• 6.4 Nonlinear Mechanical Systems
• 6.5 Chaos in Dynamical Systems
• CHAPTER 7 Laplace Transform Methods
• 7.1 Laplace Transforms and Inverse Transforms
• 7.2 Transformation of Initial Value Problems
• 7.3 Translation and Partial Fractions
• 7.4 Derivatives, Integrals, and Products of Transforms
• 7.5 Periodic and Piecewise Continuous Input Functions
• 7.6 Impulses and Delta Functions
• CHAPTER 8 Power Series Methods
• 8.1 Introduction and Review of Power Series
• 8.2 Series Solutions Near Ordinary Points
• 8.3 Regular Singular Points
• 8.4 Method of Frobenius: The Exceptional Cases
• 8.5 Bessel’s Equation
• 8.6 Applications of Bessel Functions
• CHAPTER 9 Fourier Series Methods and Partial Differential Equations
• 9.1 Periodic Functions and Trigonometric Series
• 9.2 General Fourier Series and Convergence
• 9.3 Fourier Sine and Cosine Series
• 9.4 Applications of Fourier Series
• 9.5 Heat Conduction and Separation of Variables
• 9.6 Vibrating Strings and the One-Dimensional Wave Equation
• 9.7 Steady-State Temperature and Laplace’s Equation
• CHAPTER 10 Eigenvalue Methods and Boundary Value Problems
• 10.1 Sturm–Liouville Problems and Eigenfunction Expansions
• 10.2 Applications of Eigenfunction Series
• 10.3 Steady Periodic Solutions and Natural Frequencies
• 10.4 Cylindrical Coordinate Problems
• 10.5 Higher-Dimensional Phenomena
• References for Further Study
• Appendix: Existence and Uniqueness of Solutions
• Answers to Selected Problems
• Index
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