# Ordinary Differential Equations

## An Introduction to the Fundamentals, 1st Edition

CRC Press

854 pages | 94 B/W Illus.

Hardback: 9781498733816
pub: 2015-12-08
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### Description

Ordinary Differential Equations: An Introduction to the Fundamentals is a rigorous yet remarkably accessible textbook ideal for an introductory course in ordinary differential equations. Providing a useful resource both in and out of the classroom, the text:

• Employs a unique expository style that explains the how and why of each topic covered
• Allows for a flexible presentation based on instructor preference and student ability
• Supports all claims with clear and solid proofs
• Includes material rarely found in introductory texts

Ordinary Differential Equations: An Introduction to the Fundamentals alsoincludes access to an author-maintained website featuring detailed solutions and a wealth of bonus material. Use of a math software package that can do symbolic calculations, graphing, and so forth, such as Maple or Mathematica®, is highly recommended, but not required.

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

Integration and Differential Equations

Directly-Integrable Equations

On Using Indefinite Integrals

On Using Definite Integrals

Integrals of Piecewise-Defined Functions

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

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

Linear First-Order Equations

Basic Notions

Solving First-Order Linear Equations

On Using Definite Integrals with Linear Equations

Integrability, Existence and Uniqueness

Simplifying Through Substitution

Basic Notions

Linear Substitutions

Homogeneous Equations

Bernoulli Equations

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

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

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

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

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

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

General Solutions to Homogeneous Linear Differential Equations

Second-Order Equations (Mainly)

Homogeneous Linear Equations of Arbitrary Order

Linear Independence and Wronskians

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

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

Springs: Part I

Modeling the Action

The Mass/Spring Equation and Its Solutions

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

Euler Equations

Second-Order Euler Equations

The Special Cases

Euler Equations of Any Order

The Relation between Euler and Constant Coefficient Equations

Nonhomogeneous Equations in General

General Solutions to Nonhomogeneous Equations

Superposition for Nonhomogeneous Equations

Reduction of Order

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

Springs: Part II

The Mass/Spring System

Constant Force

Resonance and Sinusoidal Forces

More on Undamped Motion under Nonresonant Sinusoidal Forces

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

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

Differentiation and the Laplace Transform

Transforms of Derivatives

Derivatives of Transforms

Transforms of Integrals and Integrals of Transforms

Appendix: Differentiating the Transform

The Inverse Laplace Transform

Basic Notions

Linearity and Using Partial Fractions

Inverse Transforms of Shifted Functions

Convolution

Convolution, the Basics

Convolution and Products of Transforms

Convolution and Differential Equations (Duhamel’s Principle)

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

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

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

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

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

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

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

Validating the Method of Frobenius

Basic Assumptions and Symbology

The Indicial Equation and Basic Recursion Formula

The Easily Obtained Series Solutions

Second Solutions When r1 = r2

Second Solutions When r1 – r2 = 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

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

Appendix: Author’s Guide to Using This Text

Overview

Chapter-by-Chapter Guide

Kenneth B. Howell earned bachelor degrees in both mathematics and physics from Rose-Hulman Institute of Technology, and master’s and doctoral degrees in mathematics from Indiana University. For more than thirty years, he was a professor in the Department of Mathematical Sciences of the University of Alabama in Huntsville (retiring in 2014). During his academic career, Dr. Howell published numerous research articles in applied and theoretical mathematics in prestigious journals, served as a consulting research scientist for various companies and federal agencies in the space and defense industries, and received awards from the College and University for outstanding teaching. He is also the author of Principles of Fourier Analysis (Chapman & Hall/CRC, 2001).