# Fundamentals of Differential Equations, Global Edition eBook (9e)

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About the book: For one-semester sophomore- or junior-level courses in Differential Equations. An introduction to the basic theory and applications of differential equations.

**Fundamentals of Differential Equations** presents the basic theory of differential equations and offers a variety of modern applications in science and engineering. This flexible text allows instructors to adapt to various course emphases (theory, methodology, applications, and numerical methods) and to use commercially available computer software.

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##### Table of contents

Our Goal

New to This Edition

Prerequisites

Sample Syllabi

Retained Features

Technology and Supplements

Acknowledgments

Acknowledgments for the Global Edition

Chapter 1: Introduction

1.1. Background

1.2. Solutions and Initial Value Problems

1.3. Direction Fields

1.4. The Approximation Method of Euler

Chapter 1: Summary

Review Problems for Chapter 1

Technical Writing Exercises for Chapter 1

Projects for Chapter 1

A. Picard’s Method

B. The Phase Line

C. Applications to Economics

D. Taylor Series Method

Chapter 2: First-Order Differential Equations

2.1. Introduction: Motion of a Falling Body

2.2. Separable Equations

2.3. Linear Equations

2.4. Exact Equations

2.5. Special Integrating Factors

2.6. Substitutions and Transformations

Chapter 2: Summary

Review Problems for Chapter 2

Technical Writing Exercises for Chapter 2

Projects for Chapter 2

A. Oil Spill in a Canal

B. Differential Equations in Clinical Medicine

C. Torricelli’s Law of Fluid Flow

D. The Snowplow Problem

E. Two Snowplows

F. Clairaut Equations and Singular Solutions

G. Multiple Solutions of a First-Order Initial Value Problem

H. Utility Functions and Risk Aversion

I. Designing a Solar Collector

J. Asymptotic Behavior of Solutions to Linear Equations

Chapter 3: Mathematical Models and Numerical Methods Involving First-Order Equations

3.1. Mathematical Modeling

3.2. Compartmental Analysis

3.3. Heating and Cooling of Buildings

3.4. Newtonian Mechanics

3.5. Electrical Circuits

3.6. Numerical Methods: A Closer Look At Euler’s Algorithm

3.7. Higher-Order Numerical Methods: Taylor and Runge–Kutta

Projects for Chapter 3

A. Dynamics of HIV Infection

B. Aquaculture

C. Curve of Pursuit

D. Aircraft Guidance in a Crosswind

E. Market Equilibrium: Stability and Time Paths

F. Stability of Numerical Methods

G. Period Doubling and Chaos

Chapter 4: Linear Second-Order Equations

4.1. Introduction: The Mass–Spring Oscillator

4.2. Homogeneous Linear Equations: The General Solution

4.3. Auxiliary Equations with Complex Roots

4.4. Nonhomogeneous Equations: The Method of Undetermined Coefficients

4.5. The Superposition Principle and Undetermined Coefficients Revisited

4.6. Variation of Parameters

4.7. Variable-Coefficient Equations

4.8. Qualitative Considerations for Variable-Coefficient and Nonlinear Equations

4.9. A Closer Look at Free Mechanical Vibrations

4.10. A Closer Look at Forced Mechanical Vibrations

Chapter 4: Summary

Review Problems for Chapter 4

Technical Writing Exercises for Chapter 4

Projects for Chapter 4

A. Nonlinear Equations Solvable by First-Order Techniques

B. Apollo Reentry

C. Simple Pendulum

D. Linearization of Nonlinear Problems

E. Convolution Method

F. Undetermined Coefficients Using Complex Arithmetic

G. Asymptotic Behavior of Solutions

H. Gravity Train†

Chapter 5: Introduction to Systems and Phase Plane Analysis

5.1. Interconnected Fluid Tanks

5.2. Differential Operators and the Elimination Method* for Systems

5.3. Solving Systems and Higher-Order Equations Numerically

5.4. Introduction to the Phase Plane

5.5. Applications to Biomathematics: Epidemic and Tumor Growth Models

5.6. Coupled Mass–Spring Systems

5.7. Electrical Systems

5.8. Dynamical Systems, Poincaré Maps, and Chaos

Chapter 5: Summary

Review Problems for Chapter 5

Projects for Chapter 5

A. Designing a Landing System for Interplanetary Travel

B. Spread of Staph Infections in Hospitals—Part I

C. Things That Bob

D. Hamiltonian Systems

E. Cleaning Up the Great Lakes

F. The 2014-2015 Ebola Epidemic

G. Phase-Locked Loops

Chapter 6: Theory of Higher-Order Linear Differential Equations

6.1. Basic Theory of Linear Differential Equations

6.2. Homogeneous Linear Equations with Constant Coefficients

6.3. Undetermined Coefficients and the Annihilator Method

6.4. Method Of Variation of Parameters

Chapter 6: Summary

Review Problems for Chapter 6

Technical Writing Exercises for Chapter 6

Projects for Chapter 6

A. Computer Algebra Systems and Exponential Shift

B. Justifying the Method of Undetermined Coefficients

C. Transverse Vibrations of a Beam

D. Higher-Order Difference Equations

Chapter 7: Laplace Transforms

7.1. Introduction: A Mixing Problem

7.2. Definition of the Laplace Transform

7.3. Properties of the Laplace Transform

7.4. Inverse Laplace Transform

7.5. Solving Initial Value Problems

7.6. Transforms of Discontinuous Functions

7.7. Transforms of Periodic and Power Functions

7.8. Convolution

7.9. Impulses and the Dirac Delta Function

7.10. Solving Linear Systems with Laplace Transforms

Chapter 7: Summary

Review Problems for Chapter 7

Technical Writing Exercises for Chapter 7

Projects for Chapter 7

A. Duhamel’s Formulas

B. Frequency Response Modeling

C. Determining System Parameters

Chapter 8: Series Solutions of Differential Equations

8.1. Introduction: The Taylor Polynomial Approximation

8.2. Power Series and Analytic Functions

8.3. Power Series Solutions to Linear Differential Equations

8.4. Equations with Analytic Coefficients

8.5. Cauchy–Euler (Equidimensional) Equations

8.6. Method of Frobenius

8.7. Finding a Second Linearly Independent Solution

8.8. Special Functions

Chapter 8: Summary

Review Problems for Chapter 8

Technical Writing Exercises for Chapter 8

Projects for Chapter 8

A. Alphabetization Algorithms

B. Spherically Symmetric Solutions to Schrödinger’s Equation for the Hydrogen Atom

C. Airy’s Equation

D. Buckling of a Tower

E. Aging Spring and Bessel Functions

Chapter 9: Matrix Methods for Linear Systems

9.1. Introduction

9.2. Review 1: Linear Algebraic Equations

9.3. Review 2: Matrices and Vectors

9.4. Linear Systems in Normal Form

9.5. Homogeneous Linear Systems with Constant Coefficients

9.6. Complex Eigenvalues

9.7. Nonhomogeneous Linear Systems

9.8. The Matrix Exponential Function

Chapter 9: Summary

Review Problems for Chapter 9

Technical Writing Exercises for Chapter 9

Projects for Chapter 9

A. Uncoupling Normal Systems

B. Matrix Laplace Transform Method

C. Undamped Second-Order Systems

Chapter 10: Partial Differential Equations

10.1. Introduction: A Model for Heat Flow

10.2. Method of Separation of Variables

10.3. Fourier Series

10.4. Fourier Cosine and Sine Series

10.5. The Heat Equation

10.6. The Wave Equation

10.7. Laplace’s Equation

Chapter 10: Summary

Technical Writing Exercises for Chapter 10

Projects for Chapter 10

A. Steady-State Temperature Distribution in a Circular Cylinder

B. Laplace Transform Solution of the Wave Equation

C. Green’s Function

D. Numerical Method for Δu = f on α Rectangle

E. The Telegrapher’s Equation and the Cable Equation

Appendices

Appendix A: Review of Integration Techniques

Appendix B: Newton’s Method

Appendix C: Simpson’s Rule

Appendix D: Cramer’s Rule

Appendix E: Method of Least Squares

Appendix F: Runge–Kutta Procedure for n Equations

Appendix G: Software for Analyzing Differential Equations

Answers to Odd-Numbered Problems

Index

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