Chapter 1. First-Order Differential Equations
1.1 Differential Equations and Mathematical Models
1.2 Geometric Analysis and Existence/Uniqueness
1.3 Separable Equations & Applications
1.4 Linear Equations & Applications
1.5 Additional Methods
1.6 Additional Exercises

Chapter 2. Second-order Differential Equations
2.1 Introduction to Higher-Order Equations
2.2 General Solutions for Second-Order Equations
2.3 Homogeneous Equations with Constant Coefficients
2.4 Free Mechanical Vibrations
2.5 Nonhomogeneous Equations with Constant Coefficients
2.6 Forced Mechanical Vibrations
2.7 Electrical Circuits
2.8 Additional Exercises

Chapter 3. Laplace Transform
3.1 Laplace Transform and Its Inverse
3.2 Transforms of Derivatives and Initial Value Problems
3.3 Shifting Theorems
3.4 Discontinuous Inputs
3.5 Convolutions
3.6 Additional Exercises

Chapter 4. Systems of Linear Equations and Matrices
4.1 Introduction to Systems and Matrices
4.2 Gaussian Elimination
4.3 Reduced Row-Echelon Form and Rank
4.4 Inverse of a Square Matrix
4.5 The Determinant of a Square Matrix
4.6 Cofactor Expansions
4.7 Additional Exercises

Chapter 5. Vector Spaces
5.1 Vectors in Rn
5.2 General Vector Spaces
5.3 Subspaces and Spanning Sets
5.4 Linear Independence
5.5 Bases and Dimension
5.6 Row and Column Spaces
5.7 Inner Products and Orthogonality
5.8 Additional Exercises

Chapter 6. Linear Transformations and Eigenvalues
6.1 Introduction to Transformations & Eigenvalues
6.2 Diagonalization and Similarity
6.3 Symmetric and Orthogonal Matrices
6.4 Additional Exercises

Chapter 7. Systems of First-Order Equations
7.1 Introduction to First-Order Systems
7.2 Theory of First-Order Linear Systems
7.3 Eigenvalue Method for Homogeneous Systems
7.4 Applications to Multiple Tank Mixing
7.5 Applications to Mechanical Vibrations
7.6 Additional Exercises

Appendices
A. Complex Numbers
B. Review of Partial Fractions
C. Table of Integrals
D. Table of Laplace Transforms
E. Answers to Some Exercises