- Chapter 1: Introduction
- Section 1.1: Background
- Section 1.2: Solutions and Initial Value Problems
- Section 1.3: Direction Fields
- Section 1.4: The Approximation Method of Euler

- Chapter 2: First-Order Differential Equations
- Section 2.1: Introduction: Motion of a Falling Body
- Section 2.2: Separable Equations
- Section 2.3: Linear Equations
- Section 2.4: Exact Equations
- Section 2.5: Special Integrating Factors
- Section 2.6: Substitutions and Transformations

- Chapter 3: Mathematical Models and Numerical Methods Involving First-Order Equations
- Section 3.1: Mathematical Modeling
- Section 3.2: Compartmental Analysis
- Section 3.3: Heating and Cooling of Buildings
- Section 3.4: Newtonian Mechanics
- Section 3.5: Electrical Circuits
- Section 3.6: Numerical Methods: A Closer Look At Euler's Algorithm
- Section 3.7: Higher-Order Numerical Methods: Taylor and Runge--Kutta

- Chapter 4: Linear Second-Order Equations
- Section 4.1: Introduction: The Mass-Spring Oscillator
- Section 4.2: Homogeneous Linear Equations: The General Solution
- Section 4.3: Auxiliary Equations with Complex Roots
- Section 4.4: Nonhomogeneous Equations: the Method of Undetermined Coefficients
- Section 4.5: The Superposition Principle and Undetermined Coefficients Revisited
- Section 4.6: Variation of Parameters
- Section 4.7: Variable-Coefficient Equations
- Section 4.8: Qualitative Considerations for Variable-Coefficient and Nonlinear Equations
- Section 4.9: A Closer Look at Free Mechanical Vibrations
- Section 4.10: A Closer Look at Forced Mechanical Vibrations

- Chapter 5: Introduction to Systems and Phase Plane Analysis
- Section 5.1: Interconnected Fluid Tanks
- Section 5.2: Differential Operators and the Elimination Method for Systems
- Section 5.3: Solving Systems and Higher-Order Equations Numerically
- Section 5.4: Introduction to the Phase Plane
- Section 5.5: Applications to Biomathematics: Epidemic and Tumor Growth Models
- Section 5.6: Coupled Mass-Spring Systems
- Section 5.7: Electrical Systems
- Section 5.8: Dynamical Systems, Poincare Maps, and Chaos

- Chapter 6: Theory of Higher-Order Linear Differential Equations
- Section 6.1: Basic Theory of Linear Differential Equations
- Section 6.2: Homogeneous Linear Equations with Constant Coefficients
- Section 6.3: Undetermined Coefficients and the Annihilator Method
- Section 6.4: Method Of Variation of Parameters

- Chapter 7: Laplace Transforms
- Section 7.1: Introduction: A Mixing Problem
- Section 7.2: Definition of the Laplace Transform
- Section 7.3: Properties of the Laplace Transform
- Section 7.4: Inverse Laplace Transform
- Section 7.5: Solving Initial Value Problems
- Section 7.6: Transforms of Discontinuous Functions
- Section 7.7: Transforms of Periodic and Power Functions
- Section 7.8: Convolution
- Section 7.9: Impulses and the Dirac Delta Function
- Section 7.10: Solving Linear Systems with Laplace Transforms

- Chapter 8: Series Solutions of Differential Equations
- Section 8.1: Introduction: The Taylor Polynomial Approximation
- Section 8.2: Power Series and Analytic Functions
- Section 8.3: Power Series Solutions to Linear Differential Equations
- Section 8.4: Equations with Analytic Coefficients
- Section 8.5: Cauchyâ€“Euler (Equidimensional) Equations
- Section 8.6: Method of Frobenius
- Section 8.7: Finding a Second Linearly Independent Solution
- Section 8.8: Special Functions

- Chapter 9: Matrix Methods for Linear Systems
- Section 9.1: Introduction
- Section 9.2: Review 1: Linear Algebraic Equations
- Section 9.3: Review 2: Matrices and Vectors
- Section 9.4: Linear Systems in Normal Form
- Section 9.5: Homogeneous Linear Systems with Constant Coefficients
- Section 9.6: Complex Eigenvalues
- Section 9.7: Nonhomogeneous Linear Systems
- Section 9.8: The Matrix Exponential Function

- Chapter 10: Partial Differential Equations
- Section 10.1: Introduction: A Model for Heat Flow
- Section 10.2: Method of Separation of Variables
- Section 10.3: Fourier Series
- Section 10.4: Fourier Cosine and Sine Series
- Section 10.5: The Heat Equation
- Section 10.6: The Wave Equation
- Section 10.7: Laplace's Equation

Exercises 1.1.1 -