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Differential Equations: An Introduction to Modern Methods and Applications, 2nd Edition

Book Description

The modern landscape of technology and industry demands an equally modern approach to differential equations in the classroom. Designed for a first course in differential equations, the second edition of Brannan/Boyce's Differential Equations: An Introduction to Modern Methods and Applications is consistent with the way engineers and scientists use mathematics in their daily work. The focus on fundamental skills, careful application of technology, and practice in modeling complex systems prepares students for the realities of the new millennium, providing the building blocks to be successful problem-solvers in today's workplace.

The text emphasizes a systems approach to the subject and integrates the use of modern computing technology in the context of contemporary applications from engineering and science. Section exercises throughout the text provide hands-on experience in modeling, analysis, and computer experimentation. Projects at the end of each chapter provide additional opportunities for students to explore the role played by differential equations in the sciences and engineering.

Brannan/Boyce's Differential Equations 2e is available with WileyPLUS, an online teaching and learning environment initially developed for Calculus and Differential Equations courses. WileyPLUS integrates the complete digital textbook, incorporating robust student and instructor resources with online auto-graded homework to create a singular online learning suite so powerful and effective that no course is complete without it.

Table of Contents

  1. Coverpage
  2. Titlepage
  3. Copyright
  4. Preface
  5. Contents
  6. 1 Introduction
    1. 1.1 Mathematical Models, Solutions, and Direction Fields
    2. 1.2 Linear Equations: Method of Integrating Factors
    3. 1.3 Numerical Approximations: Euler’s Method
    4. 1.4 Classification of Differential Equations
  7. 2 First Order Differential Equations
    1. 2.1 Separable Equations
    2. 2.2 Modeling with First Order Equations
    3. 2.3 Differences Between Linear and Nonlinear Equations
    4. 2.4 Autonomous Equations and Population Dynamics
    5. 2.5 Exact Equations and Integrating Factors
    6. 2.6 Accuracy of Numerical Methods
    7. 2.7 Improved Euler and Runge–Kutta Methods
    8. Projects
      1. 2.P.1 Harvesting a Renewable Resource
      2. 2.P.2 Designing a Drip Dispenser for a Hydrology Experiment
      3. 2.P.3 A Mathematical Model of a Groundwater Contaminant Source
      4. 2.P.4 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin
  8. 3 Systems of Two First Order Equations
    1. 3.1 Systems of Two Linear Algebraic Equations
    2. 3.2 Systems of Two First Order Linear Differential Equations
    3. 3.3 Homogeneous Linear Systems with Constant Coefficients
    4. 3.4 Complex Eigenvalues
    5. 3.5 Repeated Eigenvalues
    6. 3.6 A Brief Introduction to Nonlinear Systems
    7. 3.7 Numerical Methods for Systems of First Order Equations
    8. Projects
      1. 3.P.1 Eigenvalue-Placement Design of a Satellite Attitude Control System
      2. 3.P.2 Estimating Rate Constants for an Open Two-Compartment Model
      3. 3.P.3 The Ray Theory of Wave Propagation
      4. 3.P.4 A Blood-Brain Pharmacokinetic Model
  9. 4 Second Order Linear Equations
    1. 4.1 Definitions and Examples
    2. 4.2 Theory of Second Order Linear Homogeneous Equations
    3. 4.3 Linear Homogeneous Equations with Constant Coefficients
    4. 4.4 Mechanical and Electrical Vibrations
    5. 4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
    6. 4.6 Forced Vibrations, Frequency Response, and Resonance
    7. 4.7 Variation of Parameters
    8. Projects
      1. 4.P.1 A Vibration Insulation Problem
      2. 4.P.2 Linearization of a Nonlinear Mechanical System
      3. 4.P.3 A Spring-Mass Event Problem
      4. 4.P.4 Uniformly Distributing Points on a Sphere
      5. 4.P.5 Euler–Lagrange Equations
  10. 5 The Laplace Transform
    1. 5.1 Definition of the Laplace Transform
    2. 5.2 Properties of the Laplace Transform
    3. 5.3 The Inverse Laplace Transform
    4. 5.4 Solving Differential Equations with Laplace Transforms
    5. 5.5 Discontinuous Functions and Periodic Functions
    6. 5.6 Differential Equations with Discontinuous Forcing Functions
    7. 5.7 Impulse Functions
    8. 5.8 Convolution Integrals and Their Applications
    9. 5.9 Linear Systems and Feedback Control
    10. Projects
      1. 5.P.1 An Electric Circuit Problem
      2. 5.P.2 Effects of Pole Locations on Step Responses of Second Order Systems
      3. 5.P.3 The Watt Governor, Feedback Control, and Stability
  11. 6 Systems of First Order Linear Equations
    1. 6.1 Definitions and Examples
    2. 6.2 Basic Theory of First Order Linear Systems
    3. 6.3 Homogeneous Linear Systems with Constant Coefficients
    4. 6.4 Nondefective Matrices with Complex Eigenvalues
    5. 6.5 Fundamental Matrices and the Exponential of a Matrix
    6. 6.6 Nonhomogeneous Linear Systems
    7. 6.7 Defective Matrices
    8. Projects
      1. 6.P.1 A Compartment Model of Heat Flow in a Rod
      2. 6.P.2 Earthquakes and Tall Buildings
      3. 6.P.3 Controlling a Spring-Mass System to Equilibrium
  12. 7 Nonlinear Differential Equations and Stability
    1. 7.1 Autonomous Systems and Stability
    2. 7.2 Almost Linear Systems
    3. 7.3 Competing Species
    4. 7.4 Predator–Prey Equations
    5. 7.5 Periodic Solutions and Limit Cycles
    6. 7.6 Chaos and Strange Attractors: The Lorenz Equations
    7. Projects
      1. 7.P.1 Modeling of Epidemics
      2. 7.P.2 Harvesting in a Competitive Environment
      3. 7.P.3 The Rössler System
  13. A Matrices and Linear Algebra
    1. A.1 Matrices
    2. A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank
    3. A.3 Determinants and Inverses
    4. A.4 The Eigenvalue Problem
  14. B Complex Variables
  15. Answers to Selected Problems
  16. References
  17. Photo Credits
  18. Index