### Table of Contents

**THE BASICS**

**The Starting Point: Basic Concepts and Terminology**

Differential Equations: Basic Definitions and Classifications

Why Care about Differential Equations? Some Illustrative Examples

More on Solutions

Additional Exercises

**Integration and Differential Equations**

Directly-Integrable Equations

On Using Indefinite Integrals

On Using Definite Integrals

Integrals of Piecewise-Defined Functions

Additional Exercises

**FIRST-ORDER EQUATIONS**

**Some Basics about First-Order Equations**

Algebraically Solving for the Derivative

Constant (or Equilibrium) Solutions

On the Existence and Uniqueness of Solutions

Confirming the Existence of Solutions (Core Ideas)

Details in the Proof of Theorem 3.1

On Proving Theorem 3.2

Appendix: A Little Multivariable Calculus

Additional Exercises

**Separable First-Order Equations**

Basic Notions

Constant Solutions

Explicit Versus Implicit Solutions

Full Procedure for Solving Separable Equations

Existence, Uniqueness, and False Solutions

On the Nature of Solutions to Differential Equations

Using and Graphing Implicit Solutions

On Using Definite Integrals with Separable Equations

Additional Exercises

**Linear First-Order Equations**

Basic Notions

Solving First-Order Linear Equations

On Using Definite Integrals with Linear Equations

Integrability, Existence and Uniqueness

Additional Exercises

**Simplifying Through Substitution**

Basic Notions

Linear Substitutions

Homogeneous Equations

Bernoulli Equations

Additional Exercises

**The Exact Form and General Integrating Factors**

The Chain Rule

The Exact Form, Defined

Solving Equations in Exact Form

Testing for Exactness—Part I

"Exact Equations": A Summary

Converting Equations to Exact Form

Testing for Exactness—Part II

Additional Exercises

**Slope Fields: Graphing Solutions without the Solutions**

Motivation and Basic Concepts

The Basic Procedure

Observing Long-Term Behavior in Slope Fields

Problem Points in Slope Fields, and Issues of Existence and Uniqueness

Tests for Stability

Additional Exercises

**Euler’s Numerical Method**

Deriving the Steps of the Method

Computing via Euler’s Method (Illustrated)

What Can Go Wrong

Reducing the Error

Error Analysis for Euler’s Method

Additional Exercises

**The Art and Science of Modeling with First-Order Equations**

Preliminaries

A Rabbit Ranch

Exponential Growth and Decay

The Rabbit Ranch, Again

Notes on the Art and Science of Modeling

Mixing Problems

Simple Thermodynamics

Appendix: Approximations That Are Not Approximations

Additional Exercises

**SECOND- AND HIGHER-ORDER EQUATIONS**

**Higher-Order Equations: Extending First-Order Concepts**

Treating Some Second-Order Equations as First-Order

The Other Class of Second-Order Equations "Easily Reduced" to First-Order

Initial-Value Problems

On the Existence and Uniqueness of Solutions

Additional Exercises

**Higher-Order Linear Equations and the Reduction of Order Method**

Linear Differential Equations of All Orders

Introduction to the Reduction of Order Method

Reduction of Order for Homogeneous Linear Second-Order Equations

Reduction of Order for Nonhomogeneous Linear Second-Order Equations

Reduction of Order in General

Additional Exercises

**General Solutions to Homogeneous Linear Differential Equations**

Second-Order Equations (Mainly)

Homogeneous Linear Equations of Arbitrary Order

Linear Independence and Wronskians

Additional Exercises

**Verifying the Big Theorems and an Introduction to Differential Operators**

Verifying the Big Theorem on Second-Order, Homogeneous Equations

Proving the More General Theorems on General Solutions and Wronskians

Linear Differential Operators

Additional Exercises

**Second-Order Homogeneous Linear Equations with Constant Coefficients**

Deriving the Basic Approach

The Basic Approach, Summarized

Case 1: Two Distinct Real Roots

Case 2: Only One Root

Case 3: Complex Roots

Summary

Additional Exercises

**Springs: Part I**

Modeling the Action

The Mass/Spring Equation and Its Solutions

Additional Exercises

**Arbitrary Homogeneous Linear Equations with Constant Coefficients**

Some Algebra

Solving the Differential Equation

More Examples

On Verifying Theorem 17.2

On Verifying Theorem 17.3

Additional Exercises

**Euler Equations**

Second-Order Euler Equations

The Special Cases

Euler Equations of Any Order

The Relation between Euler and Constant Coefficient Equations

Additional Exercises

**Nonhomogeneous Equations in General**

General Solutions to Nonhomogeneous Equations

Superposition for Nonhomogeneous Equations

Reduction of Order

Additional Exercises

**Method of Undetermined Coefficients (aka: Method of Educated Guess)**

Basic Ideas

Good First Guesses for Various Choices of *g*

When the First Guess Fails

Method of Guess in General

Common Mistakes

Using the Principle of Superposition

On Verifying Theorem 20.1

Additional Exercises

**Springs: Part II**

The Mass/Spring System

Constant Force

Resonance and Sinusoidal Forces

More on Undamped Motion under Nonresonant Sinusoidal Forces

Additional Exercises

**Variation of Parameters (A Better Reduction of Order Method)**

Second-Order Variation of Parameters

Variation of Parameters for Even Higher Order Equations

The Variation of Parameters Formula

Additional Exercises

**THE LAPLACE TRANSFORM**

**The Laplace Transform (Intro)**

Basic Definition and Examples

Linearity and Some More Basic Transforms

Tables and a Few More Transforms

The First Translation Identity (And More Transforms)

What Is "Laplace Transformable"? (and Some Standard Terminology)

Further Notes on Piecewise Continuity and Exponential Order

Proving Theorem 23.5

Additional Exercises

**Differentiation and the Laplace Transform**

Transforms of Derivatives

Derivatives of Transforms

Transforms of Integrals and Integrals of Transforms

Appendix: Differentiating the Transform

Additional Exercises

**The Inverse Laplace Transform**

Basic Notions

Linearity and Using Partial Fractions

Inverse Transforms of Shifted Functions

Additional Exercises

**Convolution**

Convolution, the Basics

Convolution and Products of Transforms

Convolution and Differential Equations (Duhamel’s Principle)

Additional Exercises

**Piecewise-Defined Functions and Periodic Functions**

Piecewise-Defined Functions

The "Translation along the -*T* -Axis" Identity

Rectangle Functions and Transforms of More Piecewise-Defined Functions

Convolution with Piecewise-Defined Functions

Periodic Functions

An Expanded Table of Identities

Duhamel’s Principle and Resonance

Additional Exercises

**Delta Functions**

Visualizing Delta Functions

Delta Functions in Modeling

The Mathematics of Delta Functions

Delta Functions and Duhamel’s Principle

Some "Issues" with Delta Functions

Additional Exercises

**POWER SERIES AND MODIFIED POWER SERIES SOLUTIONS**

**Series Solutions: Preliminaries**

Infinite Series

Power Series and Analytic Functions

Elementary Complex Analysis

Additional Basic Material That May Be Useful

Additional Exercises

**Power Series Solutions I: Basic Computational Methods**

Basics

The Algebraic Method with First-Order Equations

Validity of the Algebraic Method for First-Order Equations

The Algebraic Method with Second-Order Equations

Validity of the Algebraic Method for Second-Order Equations

The Taylor Series Method

Appendix: Using Induction

Additional Exercises

**Power Series Solutions II: Generalizations and Theory**

Equations with Analytic Coefficients

Ordinary and Singular Points, the Radius of Analyticity, and the Reduced Form

The Reduced Forms

Existence of Power Series Solutions

Radius of Convergence for the Solution Series

Singular Points and the Radius of Convergence

Appendix: A Brief Overview of Complex Calculus

Appendix: The "Closest Singular Point"

Appendix: Singular Points and the Radius of Convergence for Solutions

Additional Exercises

**Modified Power Series Solutions and the Basic Method of Frobenius**

Euler Equations and Their Solutions

Regular and Irregular Singular Points (and the Frobenius Radius of Convergence)

The (Basic) Method of Frobenius

Basic Notes on Using the Frobenius Method

About the Indicial and Recursion Formulas

Dealing with Complex Exponents

Appendix: On Tests for Regular Singular Points

Additional Exercises

**The Big Theorem on the Frobenius Method, with Applications**

The Big Theorems

Local Behavior of Solutions: Issues

Local Behavior of Solutions: Limits at Regular Singular Points

Local Behavior: Analyticity and Singularities in Solutions

Case Study: The Legendre Equations

Finding Second Solutions Using Theorem 33.2

Additional Exercises

**Validating the Method of Frobenius**

Basic Assumptions and Symbology

The Indicial Equation and Basic Recursion Formula

The Easily Obtained Series Solutions

Second Solutions When *r*_{1} = r_{2}

Second Solutions When *r*_{1} – r_{2} = K

Convergence of the Solution Series

**SYSTEMS OF DIFFERENTIAL EQUATIONS (A BRIEF INTRODUCTION)**

**Systems of Differential Equations: A Starting Point**

Basic Terminology and Notions

A Few Illustrative Applications

Converting Differential Equations to First-Order Systems

Using Laplace Transforms to Solve Systems

Existence, Uniqueness and General Solutions for Systems

Single Nth-order Differential Equations

Additional Exercises

**Critical Points, Direction Fields and Trajectories**

The Systems of Interest and Some Basic Notation

Constant/Equilibrium Solutions

"Graphing" Standard Systems

Sketching Trajectories for Autonomous Systems

Critical Points, Stability and Long-Term Behavior

Applications

Existence and Uniqueness of Trajectories

Proving Theorem 36.2

Additional Exercises

**Appendix: Author’s Guide to Using This Text**

Overview

Chapter-by-Chapter Guide

**Answers to Selected Exercises**