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Summary

Table of Contents

Author Biographies

Features

New to the Second Edition

Computer Labs

Typos and Errata

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1.1. Introduction to First-Order Equations

1.2. Separable Differential Equations

1.3. Linear Equations

1.4. Some Physical Models Arising as Separable Equations

1.5. Exact Equations

1.6. Special Integrating Factors and Substitution Methods

Chapter 2: Geometrical and Numerical Methods for First-Order Equations

2.1. Direction Fields---the Geometry of Differential Equations

2.2. Existence and Uniqueness for First-Order Equations

2.3. First-Order Autonomous Equations---Geometrical Insight

2.4. Modeling in Population Biology

2.5. Numerical Approximation: Euler and Runge-Kutta Methods

2.6. An Introduction to Autonomous Second-Order Equations

Chapter 3: Elements of Higher-Order Linear Equations

3.1. Introduction to Higher-Order Equations

3.2. Linear Independence and the Wronskian

3.3. Reduction of Order---the Case n = 2

3.4. Numerical Considerations for nth-Order Equations

3.5. Essential Topics from Complex Variables

3.6. Homogeneous Equations with Constant Coefficients

3.7. Mechanical and Electrical Vibrations

Chapter 4: Techniques of Nonhomogeneous Higher-Order Linear Equations

4.1. Nonhomogeneous Equations

4.2. Method of Undetermined Coefficients via Superposition

4.3. Method of Undetermined Coefficients via Annihilation

4.4. Exponential Response and Complex Replacement

4.5. Variation of Parameters

4.6. Cauchy-Euler (Equidimensional) Equation

4.7. Forced Vibrations

Chapter 5: Fundamentals of Systems of Differential Equations

5.1. Useful Terminology

5.2. Gaussian Elimination

5.3. Vector Spaces and Subspaces

5.4. Eigenvalues and Eigenvectors

5.5. A General Method, Part I: Solving Systems with Real & Distinct or Complex Eigenvalues

5.6. A General Method, Part II: Solving Systems with Repeated Real Eigenvalues

5.7. Matrix Exponentials

5.8. Solving Linear Nonhomogeneous Systems of Equations

Chapter 6: Geometrical Approaches and Applications of Systems of First-Order Equations

6.1. An Introduction to the Phase Plane

6.2. Nonlinear Equations and Phase Plane Analysis

6.3. Bifurcations

6.4. Epidemiological Models

6.5. Models in Ecology

Chapter 7: Laplace Transforms

7.1. Introduction

7.2. Fundamentals of the Laplace Transform

7.3. The Inverse Laplace Transform

7.4. Translated Functions, Delta Function, and Periodic Functions

7.5. The s-Domain and Poles

7.6. Solving Linear Systems using Laplace Transforms

7.7. The Convolution

Chapter 8: Series Methods

8.1. Power Series Representations of Functions

8.2. The Power Series Method

8.3. Ordinary and Singular Points

8.4. The Method of Frobenius

8.5. Bessel Functions

Chapter 9: Boundary-Value Problems and Fourier Series

9.1. Two-Point Boundary-Value Problems

9.2. Orthogonal Functions and Fourier Series

9.3. Even, Odd, and Discontinuous Functions

9.4. Simple Eigenvalue-Eigenfunction Problems

9.5. Sturm-Liouville Theory

9.6. Generalized Fourier Series

Chapter 10: Partial Differential Equations

10.1. Separable Linear Partial Differential Equations

10.2. Heat Equation

10.3. Wave Equation

10.4. Laplace Equation

10.5. Nonhomogeneous Boundary Conditions

10.6. Non-Cartesian Coordinate Systems

Appendix A: An Introduction to MATLAB, Maple, and Mathematica

A.1. MATLAB

A.2. Maple

A.3. Mathematica

Apprendix B: Selected Topics from Linear Algebra

B.1. A Primer on Matrix Algebra

B.2. Matrix Inverses, Cramer's Rule

B.3. Linear Transformations

B.4. Coordinates and Change of Basis

Answers to Odd Problems

References

Index

A Review, Computer Labs, and Projects appear at the end of each chapter.

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- Describes analytical and numerical methods for studying ODEs, BVPs, and PDEs
- Shows students how to effectively use MATLAB, Maple, and Mathematica in practice, assuming no prior knowledge of the software packages
- Covers essential linear algebra topics, such as eigenvectors, bases, and transformations, to improve students' understanding of differential equations
- Includes numerous problems of varying levels of difficulty for applied and pure math majors as well as for engineers and other nonmathematicians
- Offers answers to most of the odd problems in the back of the book
- Contains reviews and projects at the end of each chapter
- Solutions manual available upon qualifying course adoption

- Moves the computer codes to Computer Labs at the end of each chapter, which gives professors flexibility in using the technology
- Covers linear systems in their entirety before addressing applications to nonlinear systems
- Incorporates the latest versions of MATLAB, Maple, and Mathematica
- Includes new sections on complex variables, the exponential response formula for solving nonhomogeneous equations, forced vibrations, and nondimensionalization
- Highlights new applications and modeling in many fields
- Presents exercise sets that progress in difficulty
- Contains color graphs to help students better understand crucial concepts in DEs
- Provides updated and expanded projects in each chapter
- Suitable for a first undergraduate course, the book includes all the basics necessary to prepare students for their future studies in mathematics, engineering, and the sciences. It presents the syntax from MATLAB, Maple, and Mathematica to give students a better grasp of the theory and gain more insight into real-world problems. Along with covering traditional topics, the text describes a number of modern topics, such as direction fields, phase lines, the Runge-Kutta method, and epidemiological and ecological models. It also explains concepts from linear algebra so that students acquire a thorough understanding of differential equations.

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Please locate the Version Date of the book. To do so, turn to the page in the front with the copyright (immediately preceding the dedication page). In the middle of this page, you will see a Version Date, e.g., Version Date: 20141104

Go to the appropriate typo list below.

For First Edition Typos , click on the link.

Thank you for reporting the above typos and errata. We appreciate your support!!

Send additional ones to:

Stephen Wirkus (e-mail: swirkus "at" asu "dot" edu)

or

Randall Swift (e-mail: rjswift "at" cpp "dot" edu)