## Table of Contents

**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

**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

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**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 Coefcients

3.4 Mechanical Vibrations

3.5 Nonhomogeneous Equations and Undetermined Coefcients

3.6 Forced Oscillations and Resonance

3.7 Electrical Circuits

3.8 Endpoint Problems and Eigenvalues

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**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

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**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

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**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

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**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

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**8. Power Series Methods **

8.1 Introduction and Review of PowerSeries

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

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**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

**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