Textbook

# Differential Equations with Boundary Value Problems: An Introduction to Modern Methods & Applications, 2nd Edition

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.

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1 Introduction

1.1 Mathematical Models, Solutions, and Direction Fields

1.2 Linear Equations: Method of Integrating Factors

1.3 Numerical Approximations: Euler's Method

1.4 Classification of Differential Equations

2 First Order Differential Equations

2.1 Separable Equations

2.2 Modeling with First Order Equations

2.3 Differences between Linear and Nonlinear Equations

2.4 Autonomous Equations and Population Dynamics

2.5 Exact Equations and Integrating Factors

2.6 Accuracy of Numerical Methods

2.7 Improved Euler and Runge-Kutta Methods

Projects

2.P.1 Harvesting a Renewable Resource

2.P.3 Designing a Drip Dispenser for a Hydrology Experiment

2.P.4 A Mathematical Model of a Groundwater Contaminant Source

2.P.5 Monte-Carlo Option Pricing: Pricing Financial Options by Flipping a Coin

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

3.7 Numerical Methods for Systems of First Order Equations

Projects

3.P.1 Eigenvalue Placement Design of a Satellite Attitude Control System

3.P.2 Estimating Rate Constants for an Open Two-Compartment Model

3.P.3 The Ray Theory of Wave Propagation

3.P.4 A Blood-Brain Pharmacokinetic Model

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 Uniformly Distributing Points on a Sphere

4.P.5 Euler-Lagrange Equations

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 Effects of Pole Locations on Step Responses of Second Order Systems

5.P.3 The Watt Governor, Feedback Control, and Stability

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 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 A Compartment Model of Heat Flow in a Rod

6.P.2 Earthquakes and Tall Buildings

6.P.3 Controlling a Spring-Mass System to Equilibrium

7 Nonlinear Differential Equations and Stability

7.1 Almost Linear Systems

7.2 Competing Species

7.3 Predator-Prey Equations

7.4 Periodic Solutions and Limit Cycles

7.5 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

[Chapters 8-10 in Boundary Value Problems version only]

8 Series Solutions of Second Order Equations

8.1 Review of Power Systems

8.2 Series Solutions Near an Ordinary Point, Part I

8.3 Series Solutions Near an Ordinary Point, Part II

8.4 Regular Singular Points

8.5 Series Solutions Near a Regular Singular Point, Part I

8.6 Series Solutions Near a Regular Singular Point, Part II

8.7 Bessel's Equation

Projects

8.P.1 Distraction Through a Circular Aperture

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

8.P.3 Perturbation Methods

9 Partial Differential Equations and Fourier Series

9.1 Two-Point Boundary Value Problems

9.2 Fourier Series

9.3 The Fourier Convergence Theorem

9.4 Even and Odd Functions

9.5 Separation of Variables, Heat Conduction in a Rod

9.6 Other Heat Conduction Problems

9.7 The Wave Equation, Vibrations of an Elastic String

9.8 Laplace's Equation

Projects

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

9.P.2 The Transmission Line Problem

9.P.3 Solving Poisson's Equation by Finite Differences

10 Boundary Value Problems and Sturm-Liouville Theory

10.1 The Occurrence of Two-Point Boundary Value Problems

10.2 Sturm-Liouville Boundary Value Problems

10.3 Nonhomogeneous Boundary Value Problems

10.4 Singular Sturm-Liouville Problems

10.5 Further Remarks on the Method of Separation of Variables: A Bessel Series

Expansion

10.6 Series of Orthogonal Functions: Mean Convergence

Projects

10.P.1 Dynamic Behavior of a Hanging Cable

10.P.2 Advection-Dispersion: A Model for Solute Transport in Saturated Porous Media

10.P.3 Fisher's Equation for Population Growth and Dispersion

A Matrices and Linear Algebra

A.1 Matrices

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

A.3 Determinants and Inverses

A.4 The Eigenvalue Problem

B Complex Variables

New To This Edition
Clarity of Applications: Based on the advice of first-edition users and others, the authors have reorganized some topics to make key ideas stand out more clearly, and have added applications that will motivate students by catching their interest and will help to build their skills in modeling with differential equations.

User Friendliness: The 2nd edition is designed to be more student-friendly by adjusting the level and strengthening the emphasis on applications, modeling and the use of computers.

Additional Problems and Exercises: New exercises, projects, and problems invite the student to make conjectures or reach conclusions about complex situations based on computer-generated data and graphs, rather than closed-form solutions.

Stressed Topics: The important link between linear second-order equations and linear systems of dimension two is strengthened in the second edition.

Real-World Applications: New introduction to two-dimensional systems of first-order differential equations in Chapter 3. The author demonstrates the usefulness of eigenvalues in the context of a timely application involving solar energy transfer and storage in a greenhouse.

Reorganization: Sections 4.2 through 4.4 reorganized into two sections. The sections are streamlined and simplified, with optional advanced material moved to exercise sets.

Hallmark Features
Flexible Organization: Organization of chapters, sections, and projects allows for a variety of course configurations depending on desired course goals, topics, and depth of coverage.

Numerous and Varied Problems: Throughout the text, section exercises of varying levels of difficulty give students hands-on experience in modeling, analysis, and computer experimentation.

Emphasis on Systems: Systems of first order equations, a central and unifying theme of the text, are introduced early, in Chapter 3, and are used frequently thereafter.

Linear Algebra and Matrix Methods: Two-dimensional linear algebra sufficient for the study of two first order equations, taken up in Chapter 3, is presented in Section 3.1. Linear algebra and matrix methods required for the study of linear systems of dimension n (Chapter 6) are treated in Appendix A.

Contemporary Project Applications: Optional projects at the end of Chapters 2 through 10 integrate subject matter in the context of exciting, contemporary applications in science and engineering, such as controlling the attitude of a satellite, ray theory of wave propagation, uniformly distributing points on a sphere, and vibration analysis of tall buildings.

Computing Exercises: In most cases, problems requiring computer generated solutions and graphics are indicated by an icon.

Visual Elements: In addition to a large number of illustrations and graphs within the text, physical representations of dynamical systems and interactive animations available in WileyPLUS provide students with a strong visual component to the subject.

Laplace Transforms: A detailed chapter on Laplace transforms discusses systems, discontinuous and impulsive input functions, transfer functions, feedback control systems, poles, and stability.

Control Theory: Ideas and methods from the important application area of control theory are introduced in some examples and projects, and in the last section on Laplace Transforms, all of which are optional.

Recurring Themes and Applications: Important themes and applications, such as dynamical system formulation, phase portraits, linearization, stability of equilibrium solutions, vibrating systems, and frequency response are revisited and reexamined in different applications and mathematical settings.

Chapter Summaries: A summary at the end of each chapter provides students and instructors with a birds-eye view of the most important ideas in the chapter.

## Available Versions

Differential Equations with Boundary Value Problems: An Introduction to Modern Methods & Applications, 2nd Edition
by James R. Brannan
ISBN 978-0-470-59535-0