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El. knyga: Ordinary Differential Equations: An Introduction to the Fundamentals

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(The University of Alabama in Huntsville, USA)
  • Formatas: 906 pages
  • Serija: Textbooks in Mathematics
  • Išleidimo metai: 06-Dec-2019
  • Leidėjas: CRC Press
  • Kalba: eng
  • ISBN-13: 9781000702415
Kitos knygos pagal šią temą:
Ordinary Differential Equations: An Introduction to the FundamentalsDidesnis paveikslėlis
  • Formatas: 906 pages
  • Serija: Textbooks in Mathematics
  • Išleidimo metai: 06-Dec-2019
  • Leidėjas: CRC Press
  • Kalba: eng
  • ISBN-13: 9781000702415
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The Second Edition of Ordinary Differential Equations: An Introduction to the Fundamentals builds on the successful First Edition. It is unique in its approach to motivation, precision, explanation and method. Its layered approach offers the instructor opportunity for greater flexibility in coverage and depth.

Students will appreciate the author’s approach and engaging style. Reasoning behind concepts and computations motivates readers. New topics are introduced in an easily accessible manner before being further developed later. The author emphasizes a basic understanding of the principles as well as modeling, computation procedures and the use of technology. The students will further appreciate the guides for carrying out the lengthier computational procedures with illustrative examples integrated into the discussion.

Features of the Second Edition:

  • Emphasizes motivation, a basic understanding of the mathematics, modeling and use of technology

  • A layered approach that allows for a flexible presentation based on instructor's preferences and students’ abilities

  • An instructor’s guide suggesting how the text can be applied to different courses

  • New chapters on more advanced numerical methods and systems (including the Runge-Kutta method and the numerical solution of second- and higher-order equations)

  • Many additional exercises, including two "chapters" of review exercises for first- and higher-order differential equations

    • An extensive on-line solution manual

    • About the author:

    Kenneth B. Howell

     earned bachelor’s 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. 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, Second Edition

    (Chapman & Hall/CRC, 2016).



    The Second Edition of this successful text is unique in its approach to motivation, precision, explanations and methods. Topics are introduced in a more accessible way then subsequent sections develop these further. Motivating the concepts, modeling, and technology are emphasized. An engaging writing style appeals to students.

    Preface (With Important Information for the Reader)
    I The Basics 1(34)
    1 The Starting Point: Basic Concepts and Terminology
    		3(18)
    1.1 Differential Equations: Basic Definitions and Classifications
    		3(5)
    1.2 Why Care About Differential Equations? Some Illustrative Examples
    		8(6)
    1.3 More on Solutions
    		14(3)
    Additional Exercises
    		17(4)
    2 Integration and Differential Equations
    		21(14)
    2.1 Directly-Integrable Equations
    		21(2)
    2.2 On Using Indefinite Integrals
    		23(1)
    2.3 On Using Definite Integrals
    		24(4)
    2.4 Integrals of Piecewise-Defined Functions
    		28(4)
    Additional Exercises
    		32(3)
    II First-Order Equations 35(208)
    3 Some Basics about First-Order Equations
    		37(28)
    3.1 Algebraically Solving for the Derivative
    		37(2)
    3.2 Constant (or Equilibrium) Solutions
    		39(3)
    3.3 On the Existence and Uniqueness of Solutions
    		42(2)
    3.4 Confirming the Existence of Solutions (Core Ideas)
    		44(3)
    3.5 Details in the Proof of Theorem 3.1
    		47(11)
    3.6 On Proving Theorem 3.2
    		58(1)
    3.7 Appendix: A Little Multivariable Calculus
    		59(4)
    Additional Exercises
    		63(2)
    4 Separable First-Order Equations
    		65(28)
    4.1 Basic Notions
    		65(5)
    4.2 Constant Solutions
    		70(5)
    4.3 Explicit Versus Implicit Solutions
    		75(2)
    4.4 Full Procedure for Solving Separable Equations
    		77(1)
    4.5 Existence, Uniqueness, and False Solutions
    		78(3)
    4.6 On the Nature of Solutions to Differential Equations
    		81(2)
    4.7 Using and Graphing Implicit Solutions
    		83(5)
    4.8 On Using Definite Integrals with Separable Equations
    		88(2)
    Additional Exercises
    		90(3)
    5 Linear First-Order Equations
    		93(12)
    5.1 Basic Notions
    		93(3)
    5.2 Solving First-Order Linear Equations
    		96(4)
    5.3 On Using Definite Integrals with Linear Equations
    		100(2)
    5.4 Integrability, Existence and Uniqueness
    		102(1)
    Additional Exercises
    		103(2)
    6 Simplifying Through Substitution
    		105(12)
    6.1 Basic Notions
    		105(2)
    6.2 Linear Substitutions
    		107(3)
    6.3 Homogeneous Equations
    		110(3)
    6.4 Bernoulli Equations
    		113(1)
    Additional Exercises
    		114(3)
    7 The Exact Form and General Integrating Factors
    		117(26)
    7.1 The Chain Rule
    		117(2)
    7.2 The Exact Form, Defined
    		119(2)
    7.3 Solving Equations in Exact Form
    		121(6)
    7.4 Testing for Exactness - Part I
    		127(2)
    7.5 "Exact Equations": A Summary
    		129(1)
    7.6 Converting Equations to Exact Form
    		130(7)
    7.7 Testing for Exactness - Part II
    		137(4)
    Additional Exercises
    		141(2)
    8 Review Exercises for Part of Part II
    		143(2)
    9 Slope Fields: Graphing Solutions Without the Solutions
    		145(32)
    9.1 Motivation and Basic Concepts
    		145(2)
    9.2 The Basic Procedure
    		147(5)
    9.3 Observing Long-Term Behavior in Slope Fields
    		152(6)
    9.4 Problem Points in Slope Fields, and Issues of Existence and Uniqueness
    		158(7)
    9.5 Tests for Stability
    		165(7)
    Additional Exercises
    		172(5)
    10 Numerical Methods I: The Euler Method
    		177(20)
    10.1 Deriving the Steps of the Method
    		177(3)
    10.2 Computing via the Euler Method (Illustrated)
    		180(3)
    10.3 Using the Results of the Method
    		183(2)
    10.4 Reducing the Error
    		185(2)
    10.5 Error Analysis for the Euler Method
    		187(6)
    Additional Exercises
    		193(4)
    11 The Art and Science of Modeling with First-Order Equations
    		197(24)
    11.1 Preliminaries
    		197(1)
    11.2 A Rabbit Ranch
    		198(3)
    11.3 Exponential Growth and Decay
    		201(3)
    11.4 The Rabbit Ranch, Again
    		204(3)
    11.5 Notes on the Art and Science of Modeling
    		207(4)
    11.6 Mixing Problems
    		211(3)
    11.7 Simple Thermodynamics
    		214(1)
    Additional Exercises
    		215(6)
    12 Numerical Methods II: Beyond the Euler Method
    		221(22)
    12.1 Forward and Backward Euler Methods
    		221(2)
    12.2 The Improved Euler Method
    		223(7)
    12.3 A Few Other Methods Worth Brief Discussion
    		230(2)
    12.4 The Classic Runge-Kutta Method
    		232(8)
    12.5 Some Additional Comments
    		240(1)
    Additional Exercises
    		240(3)
    III Second- and Higher-Order Equations 243(206)
    13 Higher-Order Equations: Extending First-Order Concepts
    		245(18)
    13.1 Treating Some Second-Order Equations as First-Order
    		246(4)
    13.2 The Other Class of Second-Order Equations "Easily Reduced" to First-Order
    		250(3)
    13.3 Initial-Value Problems
    		253(3)
    13.4 On the Existence and Uniqueness of Solutions
    		256(3)
    Additional Exercises
    		259(4)
    14 Higher-Order Linear Equations and the Reduction of Order Method
    		263(16)
    14.1 Linear Differential Equations of All Orders
    		263(3)
    14.2 Introduction to the Reduction of Order Method
    		266(1)
    14.3 Reduction of Order for Homogeneous Linear Second-Order Equations
    		267(5)
    14.4 Reduction of Order for Nonhomogeneous Linear Second-Order Equations
    		272(3)
    14.5 Reduction of Order in General
    		275(2)
    Additional Exercises
    		277(2)
    15 General Solutions to Homogeneous Linear Differential Equations
    		279(20)
    15.1 Second-Order Equations (Mainly)
    		279(11)
    15.2 Homogeneous Linear Equations of Arbitrary Order
    		290(1)
    15.3 Linear Independence and Wronskians
    		291(3)
    Additional Exercises
    		294(5)
    16 Verifying the Big Theorems and an Introduction to Differential Operators
    		299(18)
    16.1 Verifying the Big Theorem on Second-Order, Homogeneous Equations
    		299(7)
    16.2 Proving the More General Theorems on General Solutions and Wronskians
    		306(1)
    16.3 Linear Differential Operators
    		307(7)
    Additional Exercises
    		314(3)
    17 Second-Order Homogeneous Linear Equations with Constant Coefficients
    		317(20)
    17.1 Deriving the Basic Approach
    		317(3)
    17.2 The Basic Approach, Summarized
    		320(2)
    17.3 Case 1: Two Distinct Real Roots
    		322(1)
    17.4 Case 2: Only One Root
    		323(4)
    17.5 Case 3: Complex Roots
    		327(6)
    17.6 Summary
    		333(1)
    Additional Exercises
    		334(3)
    18 Springs: Part I
    		337(16)
    18.1 Modeling the Action
    		337(4)
    18.2 The Mass/Spring Equation and Its Solutions
    		341(9)
    Additional Exercises
    		350(3)
    19 Arbitrary Homogeneous Linear Equations with Constant Coefficients
    		353(18)
    19.1 Some Algebra
    		353(3)
    19.2 Solving the Differential Equation
    		356(4)
    19.3 More Examples
    		360(2)
    19.4 On Verifying Theorem 19.2
    		362(6)
    19.5 On Verifying Theorem 19.3
    		368(1)
    Additional Exercises
    		369(2)
    20 Euler Equations
    		371(14)
    20.1 Second-Order Euler Equations
    		371(3)
    20.2 The Special Cases
    		374(4)
    20.3 Euler Equations of Any Order
    		378(3)
    20.4 The Relation Between Euler and Constant Coefficient Equations
    		381(1)
    Additional Exercises
    		382(3)
    21 Nonhomogeneous Equations in General
    		385(10)
    21.1 General Solutions to Nonhomogeneous Equations
    		385(4)
    21.2 Superposition for Nonhomogeneous Equations
    		389(2)
    21.3 Reduction of Order
    		391(1)
    Additional Exercises
    		391(4)
    22 Method of Undetermined Coefficients (aka: Method of Educated Guess)
    		395(20)
    22.1 Basic Ideas
    		395(3)
    22.2 Good First Guesses for Various Choices of g
    		398(4)
    22.3 When the First Guess Fails
    		402(2)
    22.4 Method of Guess in General
    		404(3)
    22.5 Common Mistakes
    		407(1)
    22.6 Using the Principle of Superposition
    		408(1)
    22.7 On Verifying Theorem 22.1
    		409(3)
    Additional Exercises
    		412(3)
    23 Springs: Part II (Forced Vibrations)
    		415(16)
    23.1 The Mass/Spring System
    		415(2)
    23.2 Constant Force
    		417(1)
    23.3 Resonance and Sinusoidal Forces
    		418(6)
    23.4 More on Undamped Motion under Nonresonant Sinusoidal Forces
    		424(2)
    Additional Exercises
    		426(5)
    24 Variation of Parameters (A Better Reduction of Order Method)
    		431(16)
    24.1 Second-Order Variation of Parameters
    		431(8)
    24.2 Variation of Parameters for Even Higher Order Equations
    		439(3)
    24.3 The Variation of Parameters Formula
    		442(2)
    Additional Exercises
    		444(3)
    25 Review Exercises for Part III
    		447(2)
    IV The Laplace Transform 449(126)
    26 The Laplace Transform (Intro)
    		451(30)
    26.1 Basic Definition and Examples
    		451(6)
    26.2 Linearity and Some More Basic Transforms
    		457(2)
    26.3 Tables and a Few More Transforms
    		459(5)
    26.4 The First Translation Identity (and More Transforms)
    		464(2)
    26.5 What Is "Laplace Transformable"? (and Some Standard Terminology)
    		466(5)
    26.6 Further Notes on Piecewise Continuity and Exponential Order
    		471(3)
    26.7 Proving Theorem 26.5
    		474(3)
    Additional Exercises
    		477(4)
    27 Differentiation and the Laplace Transform
    		481(18)
    27.1 Transforms of Derivatives
    		481(5)
    27.2 Derivatives of Transforms
    		486(2)
    27.3 Transforms of Integrals and Integrals of Transforms
    		488(5)
    27.4 Appendix: Differentiating the Transform
    		493(3)
    Additional Exercises
    		496(3)
    28 The Inverse Laplace Transform
    		499(12)
    28.1 Basic Notions
    		499(2)
    28.2 Linearity and Using Partial Fractions
    		501(6)
    28.3 Inverse Transforms of Shifted Functions
    		507(2)
    Additional Exercises
    		509(2)
    29 Convolution
    		511(14)
    29.1 Convolution: The Basics
    		511(4)
    29.2 Convolution and Products of Transforms
    		515(4)
    29.3 Convolution and Differential Equations (Duhamel's Principle)
    		519(4)
    Additional Exercises
    		523(2)
    30 Piecewise-Defined Functions and Periodic Functions
    		525(32)
    30.1 Piecewise-Defined Functions
    		525(3)
    30.2 The "Translation Along the T-Axis" Identity
    		528(5)
    30.3 Rectangle Functions and Transforms of More Piecewise-Defined Functions
    		533(4)
    30.4 Convolution with Piecewise-Defined Functions
    		537(3)
    30.5 Periodic Functions
    		540(5)
    30.6 An Expanded Table of Identities
    		545(1)
    30.7 Duhamel's Principle and Resonance
    		546(7)
    Additional Exercises
    		553(4)
    31 Delta Functions
    		557(18)
    31.1 Visualizing Delta Functions
    		557(1)
    31.2 Delta Functions in Modeling
    		558(4)
    31.3 The Mathematics of Delta Functions
    		562(4)
    31.4 Delta Functions and Duhamel's Principle
    		566(2)
    31.5 Some "Issues" with Delta Functions
    		568(4)
    Additional Exercises
    		572(3)
    V Power Series and Modified Power Series Solutions 575(188)
    32 Series Solutions: Preliminaries
    		577(26)
    32.1 Infinite Series
    		577(5)
    32.2 Power Series and Analytic Functions
    		582(9)
    32.3 Elementary Complex Analysis
    		591(3)
    32.4 Additional Basic Material That May Be Useful
    		594(5)
    Additional Exercises
    		599(4)
    33 Power Series Solutions I: Basic Computational Methods
    		603(44)
    33.1 Basics
    		603(2)
    33.2 The Algebraic Method with First-Order Equations
    		605(10)
    33.3 Validity of the Algebraic Method for First-Order Equations
    		615(5)
    33.4 The Algebraic Method with Second-Order Equations
    		620(8)
    33.5 Validity of the Algebraic Method for Second-Order Equations
    		628(3)
    33.6 The Taylor Series Method
    		631(5)
    33.7 Appendix: Using Induction
    		636(5)
    Additional Exercises
    		641(6)
    34 Power Series Solutions II: Generalizations and Theory
    		647(34)
    34.1 Equations with Analytic Coefficients
    		647(1)
    34.2 Ordinary and Singular Points, the Radius of Analyticity, and the Reduced Form
    		648(4)
    34.3 The Reduced Forms
    		652(1)
    34.4 Existence of Power Series Solutions
    		653(6)
    34.5 Radius of Convergence for the Solution Series
    		659(3)
    34.6 Singular Points and the Radius of Convergence
    		662(1)
    34.7 Appendix: A Brief Overview of Complex Calculus
    		663(4)
    34.8 Appendix: The "Closest Singular Point"
    		667(4)
    34.9 Appendix: Singular Points and the Radius of Convergence for Solutions
    		671(7)
    Additional Exercises
    		678(3)
    35 Modified Power Series Solutions and the Basic Method of Frobenius
    		681(38)
    35.1 Euler Equations and Their Solutions
    		681(4)
    35.2 Regular and Irregular Singular Points (and the Frobenius Radius of Convergence)
    		685(5)
    35.3 The (Basic) Method of Frobenius
    		690(12)
    35.4 Basic Notes on Using the Frobenius Method
    		702(3)
    35.5 About the Indicial and Recursion Formulas
    		705(7)
    35.6 Dealing with Complex Exponents
    		712(1)
    35.7 Appendix: On Tests for Regular Singular Points
    		713(2)
    Additional Exercises
    		715(4)
    36 The Big Theorem on the Frobenius Method, with Applications
    		719(24)
    36.1 The Big Theorems
    		719(4)
    36.2 Local Behavior of Solutions: Issues
    		723(1)
    36.3 Local Behavior of Solutions: Limits at Regular Singular Points
    		724(3)
    36.4 Local Behavior: Analyticity and Singularities in Solutions
    		727(3)
    36.5 Case Study: The Legendre Equations
    		730(4)
    36.6 Finding Second Solutions Using Theorem 36.2
    		734(5)
    Additional Exercises
    		739(4)
    37 Validating the Method of Frobenius
    		743(20)
    37.1 Basic Assumptions and Symbology
    		743(1)
    37.2 The Indicial Equation and Basic Recursion Formula
    		744(4)
    37.3 The Easily Obtained Series Solutions
    		748(3)
    37.4 Second Solutions When 7-2 = rl
    		751(3)
    37.5 Second Solutions When ri - r2 > = K
    		754(7)
    37.6 Convergence of the Solution Series
    		761(2)
    VI Systems of Differential Equations (A Brief Introduction) 763(76)
    38 Systems of Differential Equations: A Starting Point
    		765(24)
    38.1 Basic Terminology and Notions
    		765(4)
    38.2 A Few Illustrative Applications
    		769(4)
    38.3 Converting Differential Equations to First-Order Systems
    		773(4)
    38.4 Using Laplace Transforms to Solve Systems
    		777(2)
    38.5 Existence, Uniqueness and General Solutions for Systems
    		779(4)
    38.6 Single Nth-Order Differential Equations
    		783(3)
    Additional Exercises
    		786(3)
    39 Critical Points, Direction Fields and Trajectories
    		789(32)
    39.1 The Systems of Interest and Some Basic Notation
    		789(2)
    39.2 Constant/Equilibrium Solutions
    		791(2)
    39.3 "Graphing" Standard Systems
    		793(2)
    39.4 Sketching Trajectories for Autonomous Systems
    		795(5)
    39.5 Critical Points, Stability and Long-Term Behavior
    		800(3)
    39.6 Applications
    		803(6)
    39.7 Existence and Uniqueness of Trajectories
    		809(2)
    39.8 Proving Theorem 39.2
    		811(4)
    Additional Exercises
    		815(6)
    40 Numerical Methods III: Systems and Higher-Order Equations
    		821(18)
    40.1 Brief Review of the Basic Euler Method
    		821(1)
    40.2 The Euler Method for First-Order Systems
    		822(7)
    40.3 Extending Euler's Method to Second-Order Differential Equations
    		829(7)
    Additional Exercises
    		836(3)
    Appendix: Author's Guide to Using This Text 839(10)
    A.1 Overview
    		839(1)
    A.2
    Chapter-by-Chapter Guide
    		840(9)
    Answers to Selected Exercises 849(38)
    Index 887
    Kenneth B. Howell earned bachelor degrees in both mathematics and physics from Rose-Hulman Institute of Technology, and masters 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).
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