Schaum's Outline of Differential Equations, Fifth Edition

by Richard Bronson, Gabriel B. Costa

Overview

Study smarter and stay on top of your differential equations course with the bestselling Schaum’s Outline—now with the NEW Schaum’s app and website!

Schaum’s Outline of Differential Equations, Fifth Edition is the go-to study guide for all students of science who need to learn or refresh their knowledge of differential equations. With an outline format that facilitates quick and easy review and mirrors the course in scope and sequence, this book helps you understand basic concepts and get the extra practice you need to excel in the course. It supports the all major differential equations textbooks and is useful for study in Calculus (I, II, and III), Mathematical Modeling, Introductory Differential Equations and Differential Equations.

Chapters include an Introduction to Modeling and Qualitative Methods, Classifications of First-Order Differential Equations, Linear Differential Equations, Variation of Parameters, Initial-Value Problems for Linear Differential Equations, Graphical and Numerical Methods for Solving First-Order Differential Equations, Solutions of Linear Differential Equations with Constant Coefficients by Laplace Transforms, and more.

Features:

• NEW to this edition: the new Schaum’s app and website!
• NEW CHAPTERS include Autonomous Differential Equations and Qualitative Methods; Eigenvalues and Eigenvectors; three chapters dealing with Solutions of Systems of Autonomous Equations via Eigenvalues and Eigenvectors (real and distinct, real and equal, and complex conjugate Eigenvalues)
• 20 problem-solving videos online
• 563 solved problems
• Outline format provides a quick and easy review of differential equations
• Clear, concise explanations of differential equations concepts
• Hundreds of examples with explanations of key concepts
• Supports all major textbooks for differential equations courses
• Appropriate for the following courses: Calculus (I, II, and III), Mathematical Modeling, Introductory Differential Equations, and Differential Equations

Product Details

ISBN-13:	9781264258833
Publisher:	McGraw Hill LLC
Publication date:	11/12/2021
Sold by:	Barnes & Noble
Format:	eBook
Pages:	416
File size:	57 MB
Note:	This product may take a few minutes to download.

About the Author

Richard Bronson, Ph.D., is a professor of mathematics at Farleigh Dickenson University. Bronson has served as an associate editor of the journal Simulation, as a contributing editor to SIAM News, and as a consultant to Bell Laboratories. He has conducted joint research in mathematical modeling and computer simulation at Technion-Israel Institute of Technology and the Wharton School of Business at the University of Pennsylvania. Bronson has published over 30 technical articles and books, including Schaum’s Outline of Matrix Operations and Shaum’s Outline of Operations Research.

Gabriel B. Costa, Ph.D., is an associate professor of mathematical sciences at the United States Military Academy at West Point, where he also functions as an associate chaplain. In addition to differential equations, his academic interests include mathematics equation and sabermetrics, the search for objective knowledge about baseball.

Table of Contents

Chapter 1 Basic Concepts 1

Differential Equations 1

Notation 2

Solutions 2

Initial-Value and Boundary-Value Problems 2

Chapter 2 An Introduction to Modeling and Qualitative Methods 9

Mathematical Models 9

The "Modeling Cycle" 9

Qualitative Methods 10

Chapter 3 Classifications of First-Order Differential Equations 14

Standard Form and Differential Form 14

Linear Equations 14

Bernoulli Equations 14

Homogeneous Equations 15

Separable Equations 15

Exact Equations 15

Chapter 4 Separable First-Order Differntial Equations 21

General Solution 21

Solutions to the Initial-Value Problem 21

Reduction of Homogeneous Equations 22

Chapter 5 Exact First-Order Differential Equations 31

Defining Properties 31

Method of Solution 31

Integrating Factors 32

Chapter 6 Linear First-Order Differential Equation 42

Method of Solution 42

Reduction of Bernoulli Equations 42

Chapter 7 Applications of First-Order Differential Equations 50

Growth and Decay Problems 50

Temperature Problems 50

Falling Body Problems 51

Dilution Problems 52

Electrical Circuits 52

Orthogonal Trajectories 53

Chapter 8 Linear Differential Equations: Theory of Solutions 73

Linear Differential Equations 73

Linearly Independent Solutions 74

The Wronskian 74

Nonhomogeneous Equations 74

Chapter 9 Second-Order Linear Homogeneous Differential Equations with Constant Coefficients 83

Introductory Remark 83

The Characteristic Equation 83

The General Solution 84

Chapter 10 nth-Order Linear Homogeneous Differential Equations with Constant Coefficients 89

The Characteristic Equation 89

The General Solution 90

Chapter 11 The Method of Undetermined Coefficients 94

Simple Form of the Method 94

Generalizations 95

Modifications 95

Limitations of the Method 95

Chapter 12 Variation of Parameters 103

The Method 103

Scope of the Method 104

Chapter 13 Initial-Value Problems for Linear Differential Equations 110

Chapter 14 Applications of Second-Order Linear Differential Equations 114

Spring Problems 114

Electrical Circuit Problems 115

Buoyancy Problems 116

Classifying Solutions 117

Chapter 15 Matrices 131

Matrices and Vectors 131

Matrix Addition 131

Scalar and Matrix Multiplication 132

Powers of a Square Matrix 132

Differentiation and Integration of Matrices 132

The Characteristic Equation 133

Chapter 16 e^At 140

Definition 140

Computation of e^At 140

Chapter 17 Reduction of Linear Differential Equations to a System of First-Order Equations 148

An Example 148

Reduction of an n^th Order Equation 149

Reduction of a System 150

Chapter 18 Graphical and Numerical Methods for Solving First-Order Differential Equations 157

Qualitative Methods 157

Direction Fields 157

Euler's Method 158

Stability 158

Chapter 19 Further Numerical Methods for Solving First-Order Differential Equations 176

General Remarks 176

Modified Euler's Method 177

Runge-Kutta Method 177

Adams-Bashford-Moulton Method 177

Milne's Method 177

Starting Values 178

Order of a Numerical Method 178

Chapter 20 Numerical Methods for Solving Second-Order Differential Equations Via Systems 195

Second-Order Differential Equations 195

Euler's Method 196

Runge-Kutta Method 196

Adams-Bashford-Moulton Method 196

Chapter 21 The Laplace Transform 211

Definition 211

Properties of Laplace Transforms 211

Functions of Other Independent Variables 212

Chapter 22 Invere Laplace Transforms 224

Definition 224

Manipulating Denominators 224

Manipulating Numerators 225

Chapter 23 Convolutions and the Unit Step Function 233

Convolutions 233

Unit Step Function 233

Translations 234

Chapter 24 Solutions of Linear Differential Equations with Constant Coefficients by Laplace Transforms 242

Laplace Transforms of Derivatives 242

Solutions of Differential Equations 243

Chapter 25 Solutions of Linear Systems by Laplace Transforms 249

The Method 249

Chapter 26 Solutions of Linear Differential Equations with Constant Coefficients by Matrix Methods 254

Solution of the Initial-Value Problem 254

Solution with No Initial Conditions 255

Chapter 27 Power Series Solutions of Linear Differential Equations with Variable Coefficients 262

Second-Order Equations 262

Analytic Functions and Ordinary Points 262

Solutions Around the Origin of Homogeneous Equations 263

Solutions Around the Origin of Nonhomogeneous Equations 263

Initial-Value Problems 264

Solutions Around Other Points 264

Chapter 28 Series Solutions Near a Regular Singular Point 275

Regular Singular Points 275

Method of Frobenius 275

General Solution 276

Chapter 29 Some Classical Differential Equations 290

Classical Differential Equations 290

Polynomial Solutions and Associated Concepts 290

Chapter 30 Gamma and Bessel Functions 295

Gamma Function 295

Bessel Functions 295

Algebraic Operations on Infinite Series 296

Chapter 31 An Introduction to Partial Differential Equations 304

Introductory Concepts 304

Solutions and Solution Techniques 305

Chapter 32 Second-Order Boundary-Value Problems 309

Standard Form 309

Solutions 310

Eigenvalue Problems 310

Sturm-Liouville Problems 310

Properties of Sturm-Liouville Problems 310

Chapter 33 Eigenfunction Expansions 318

Piecewise Smooth Functions 318

Fourier Sine Series 319

Fourier Cosine Series 319

Chapter 34 An Introduction to Difference Equations 325

Introduction 325

Classifications 325

Solutions 326

Chapter 35 Solving Differential Equations Using Mathematica 330

Introduction 330

Chapter 36 Solving Systems of Differential Equations via Eigenvalues Using Mathematica 334

Introduction 334

Terminology 334

Chapter 37 Qualitative Methods 339

Introduction 339

Terminology 339

Chapter 38 Euler's Method Using Microsoft Excel® 348

The Method 348

Chapter 39 Some Interesting Modeling Problems 352

Examples 352

Appendix Laplace Transforms 359

Answers to Supplementary Problems 365

Index 411