### 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

**3. Linear Systems and Matrices**

3.1 Introduction to Linear Systems

3.2 Matrices and Gaussian Elimination

3.3 Reduced Row-Echelon Matrices

3.4 Matrix Operations

3.5 Inverses of Matrices

3.6 Determinants

3.7 Linear Equations and Curve Fitting

**4. Vector Spaces**

4.1 The Vector Space R3

4.2 The Vector Space Rn and Subspaces

4.3 Linear Combinations and Independence of Vectors

4.4 Bases and Dimension for Vector Spaces

4.5 Row and Column Spaces

4.6 Orthogonal Vectors in Rn

4.7 General Vector Spaces

**5. Higher-Order Linear Differential Equations**

5.1 Introduction: Second-Order Linear Equations

5.2 General Solutions of Linear Equations

5.3 Homogeneous Equations with Constant Coefficients

5.4 Mechanical Vibrations

5.5 Nonhomogeneous Equations and Undetermined Coefficients

5.6 Forced Oscillations and Resonance

**6. Eigenvalues and Eigenvectors**

6.1 Introduction to Eigenvalues

6.2 Diagonalization of Matrices

6.3 Applications Involving Powers of Matrices

**7. Linear Systems of Differential Equations**

7.1 First-Order Systems and Applications

7.2 Matrices and Linear Systems

7.3 The Eigenvalue Method for Linear Systems

7.4 A Gallery of Solution Curves of Linear Systems

7.5 Second-Order Systems and Mechanical Applications

7.6 Multiple Eigenvalue Solutions

7.7 Numerical Methods for Systems

**8. Matrix Exponential Methods**

8.1 Matrix Exponentials and Linear Systems

8.2 Nonhomogeneous Linear Systems

8.3 Spectral Decomposition Methods

9. Nonlinear Systems and Phenomena

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