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Index
Cover
Title Page
Copyright Page
Dedication
Contents
Preface
Chapter 1: Introduction to Differential Equations
Prologue
1.1 Basic Definitions and Concepts
1.2 Some Basic Theory
Chapter 2: First-Order Differential Equations
2.1 First-Order Linear Equations
2.2 Separable Equations
2.3 Growth and Decay Phenomena
2.4 Mixing Phenomena
2.5 Cooling and Heating Phenomena
2.6 More Applications
2.7 The Direction Field and Euler’s Method
2.8 Higher-Order Numerical Methods
Chapter 3: Second-Order Linear Equations
3.1 Introduction to Second-Order Linear Equations
3.2 Fundamental Solutions of the Homogeneous Equation
3.3 Reduction of Order
3.4 Homogeneous Equations with Constant Coefficients: Real Roots
3.5 Homogeneous Equations with Constant Coefficients: Complex Roots
3.6 Nonhomogeneous Equations
3.7 Solving Nonhomogeneous Equations: Method of Undetermined Coefficients
3.8 Solving Nonhomogeneous Equations: Method of Variation of Parameters
3.9 Mechanical Systems and Simple Harmonic Motion
3.10 Unforced Damped Vibrations
3.11 Forced Vibrations
3.12 Introduction to Higher-Order Equations (Optional)
Chapter 4: Series Solutions
4.1 Introduction: A Review of Power Series
4.2 Power Series Expansions about Ordinary Points: Part I
4.3 Power Series Expansions about Ordinary Points: Part II
4.4 Series Solutions about Singular Points: The Method of Frobenius
4.5 Bessel Functions
Chapter 5: The Laplace Transform
5.1 Definition of the Laplace Transform
5.2 Properties of the Laplace Transform
5.3 The Inverse Laplace Transform
5.4 Initial-Value Problems
5.5 Step Functions and Delayed Functions
5.6 Differential Equations with Discontinuous Forcing Functions
5.7 Impulse Forcing Functions
5.8 The Convolution Integral
Chapter 6: Systems of Differential Equations
6.1 Introduction to Linear Systems: The Method of Elimination
6.2 Review of Matrices
6.3 Basic Theory of First-Order Linear Systems
6.4 Homogeneous Linear Systems with Real Eigenvalues
6.5 Homogeneous Linear Systems with Complex Eigenvalues
6.6 Nonhomogeneous Linear Systems
6.7 Nonhomogeneous Linear Systems: Laplace Transform (Optional)
6.8 Applications of Linear Systems
6.9 Numerical Solution of Systems of Differential Equations
Chapter 7: Difference Equations
7.1 Introduction to Difference Equations
7.2 Homogeneous Equations
7.3 Nonhomogeneous Equations
7.4 Applications of Difference Equations
7.5 The Logistic Equation and the Path to Chaos
7.6 Iterative Systems: Julia Sets and the Mandelbrot Set (Optional)
Chapter 8: Nonlinear Differential Equations and Chaos
8.1 Phase Plane Analysis of Autonomous Systems
8.2 Equilibrium Points and Stability for Linear Systems
8.3 Stability: Almost Linear Systems
8.4 Chaos, Poincare Sections and Strange Attractors
Chapter 9: Partial Differential Equations
9.1 Fourier Series
9.2 Fourier Sine and Cosine Series
9.3 Introduction to Partial Differential Equations
9.4 The Vibrating String: Separation of Variables
9.5 Superposition Interpretation of the Vibrating String
9.6 The Heat Equation and Separation of Variables
9.7 Laplace’s Equation Inside a Circle
Appendix: Complex Numbers and Complex-Valued Functions
Answers to Problems
Index
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