# Differential Equations: An Introduction to Modern Methods and Applications, 3rd Edition

# Differential Equations: An Introduction to Modern Methods and Applications, 3rd Edition

ISBN: 978-1-118-53177-8

Jul 2015

688 pages

## Description

Brannan/Boyce’s * Differential Equations: An Introduction to Modern Methods and Applications, 3rd Edition* is consistent with the way engineers and scientists use mathematics in their daily work. 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. 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. 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.

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**Chapter 1: Introduction**

1.1 Mathematical Models and Solutions

1.2 Qualitative Methods: Phase Lines and Direction Fields

1.3 Definitions, Classification, and Terminology

**Chapter 2: First Order Differential Equations**

2.1 Separable Equations

2.2 Linear Equations: Method of Integrating Factors

2.3 Modeling with First Order Equations

2.4 Differences Between Linear and Nonlinear Equations

2.5 Autonomous Equations and Population Dynamics

2.6 Exact Equations and Integrating Factors

2.7 Substitution Methods

Projects

2.P.1 Harvesting a Renewable Resource

2.P.2 A Mathematical Model of a Groundwater Contaminant Source

2.P.3 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin

**Chapter 3: Systems of Two First Order Equations**

3.1 Systems of Two Linear Algebraic Equations

3.2 Systems of Two First Order Linear Differential Equations

3.3 Homogeneous Linear Systems with Constant Coefficients

3.4 Complex Eigenvalues

3.5 Repeated Eigenvalues

3.6 A Brief Introduction to Nonlinear Systems

Projects

3.P.1 Estimating Rate Constants for an Open Two-Compartment Model

3.P.2 A Blood-Brain Pharmacokinetic Model

**Chapter 4: Second Order Linear Equations**

4.1 Definitions and Examples

4.2 Theory of Second Order Linear Homogeneous Equations

4.3 Linear Homogeneous Equations with Constant Coefficients

4.4 Mechanical and Electrical Vibrations

4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

4.6 Forced Vibrations, Frequency Response, and Resonance

4.7 Variation of Parameters

Projects

4.P.1 A Vibration Insulation Problem

4.P.2 Linearization of a Nonlinear Mechanical System

4.P.3 A Spring-Mass Event Problem

4.P.4 Euler-Lagrange Equations

**Chapter 5: The Laplace Transform**

5.1 Definition of the Laplace Transform

5.2 Properties of the Laplace Transform

5.3 The Inverse Laplace Transform

5.4 Solving Differential Equations with Laplace Transforms

5.5 Discontinuous Functions and Periodic Functions

5.6 Differential Equations with Discontinuous Forcing Functions

5.7 Impulse Functions

5.8 Convolution Integrals and Their Applications

5.9 Linear Systems and Feedback Control

Projects

5.P.1 An Electric Circuit Problem

5.P.2 The Watt Governor, Feedback Control, and Stability

**Chapter 6: Systems of First Order Linear Equations**

6.1 Definitions and Examples

6.2 Basic Theory of First Order Linear Systems

6.3 Homogeneous Linear Systems with Constant Coefficients

6.4 Nondefective Matrices with Complex Eigenvalues

6.5 Fundamental Matrices and the Exponential of a Matrix

6.6 Nonhomogeneous Linear Systems

6.7 Defective Matrices

Projects

6.P.1 Earthquakes and Tall Buildings

6.P.2 Controlling a Spring-Mass System to Equilibrium

**Chapter 7: Nonlinear Differential Equations and Stability**

7.1 Autonomous Systems and Stability

7.2 Almost Linear Systems

7.3 Competing Species

7.4 Predator-Prey Equations

7.5 Periodic Solutions and Limit Cycles

7.6 Chaos and Strange Attractors: The Lorenz Equations

Projects

7.P.1 Modeling of Epidemics

7.P.2 Harvesting in a Competitive Environment

7.P.3 The Rossler System

**Chapter 8: Numerical Methods**

8.1 Numerical Approximations: Euler’s Method

8.2 Accuracy of Numerical Methods

8.3 Improved Euler and Runge-Kutta Methods

8.4 Numerical Methods for Systems of First Order Equations

Projects

8.P.1 Designing a Drip Dispenser for a Hydrology Experiment

8.P.2 Monte Carlo Option Pricing: Pricing Financial Option by Flipping a Coin

**Chapter 9: Series Solutions of Second order Equations**

9.1 Review of Power Series

9.2 Series Solutions Near an Ordinary Point, Part I

9.3 Series Solutions Near an Ordinary Point, Part II

9.4 Regular Singular Points

9.5 Series Solutions Near a Regular Singular Point, Part I

9.6 Series Solutions Near a Regular Singular Point, Part II

9.7 Bessel’s Equation

Projects

9.P.1 Diffraction Through a Circular Aperature

9.P.2 Hermite Polynomials and the Quantum Mechanical Harmonic Oscillator

9.P.3 Perturbation Methods

**Chapter 10: Orthogonal Functions, Fourier Series and Boundary-Value Problems**

10.1 Orthogonal Families in the Space PC [a,b]

10.2 Fourier Series

10.3 Elementary Two-Point Boundary Value Problems

10.4 General Sturm-Liouville Boundary Value Problems

10.5 Generalized Fourier Series and Eigenfunction Expansions

10.6 Singular Boundary Value Problems

10.7 Convergence Issues

**Chapter 11: Elementary Partial Differential Equations**

11.1 Terminology

11.2 Heat Conduction in a Rod—Homogeneous Case

11.3 Heat Conduction in a Rod—Nonhomogeneous Case

11.4 Wave Equation—Vibrations of an Elastic String

11.5 Wave Equation—Vibrations of a Circular Membrane

11.6 Laplace Equation

Projects

11.P.1 Estimating the Diffusion Coefficient in the Heat Equation

11.P.2 The Transmission Line Problem

11.P.3 Solving Poisson’s Equation by Finite Differences

11.P.4 Dynamic Behavior of a Hanging Cable

11.P.5 Advection Dispersion: A Model for Solute Transport in Saturated Porous Media

11.P.6 Fisher’s Equation for Population Growth and Dispersion

**Appendices**

11.A Derivation of the Heat Equation

11.B Derivation of the Wave Equation

A: Matrices and Linear Algebra

A.1 Matrices

A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank

A.3 Determinates and Inverses

A.4 The Eigenvalue Problem

B: Complex Variables

**Answers to Selected Problems**

**References**

**Index**

- Presents more important results, theorems, and definitions in colored summary boxes to enable students to more easily review for tests and exams.
- Some topics that appear only in exercises in the second edition will be included in the main text of the third. Important examples are the substitution methods for solving homogeneous equations and Bernoulli equations.
- Numerical methods collected in a new, optional, chapter 8. The first three sections of this chapter will be accessible to students after Chapter 2.
- Places more emphasis (via discussion, examples, and problems) on how models and applications depend on parameter values.

**Prepare & Present**

**Course Materials**tohelp you personalize lessons and optimize your time, including:- PowerPoint Lecture Slides
- Instructor's Solution Manual

**Ready, Study & Practice**

**Complete online version of the textbook included****Relevant, student study tools and learning resources**to ensure positive learning outcomes, including:- eBook
- Project Activities
- Maple, Mathematica and MatLab Data Files
- Student Solutions Manual
**Immediate feedback**toboost confidence and help students see a return on investment for each study session:**Algorithmic**End-of-Section and End-of-Chapter homework questions

**Assignment**

**Pre-created activities**to encourage learning outside of the classroom, including:- Gradable Reading Assignment Questions (embedded with online text)
- Question Assignments: all end-of-chapter problems coded algorithmically with hints, links to text, whiteboard/show work feature and instructor-controlled problem solving help.
*WileyPLUS***Quickstart assignments**and presentations for the entire course created by a subject-matter expert.