Differential Equations and Boundary Value Problems: Computing and Modeling, 2/e
C. Henry Edwards
David E. Penney, both of the University of Georgia
Published April, 1999 by Prentice Hall Engineering/Science/Mathematics
Copyright 2000, 787 pp.
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For introductory courses in Differential Equations.
This text provides the conceptual development and geometric
visualization of a modern differential equations course while maintaining
the solid foundation of algebraic techniques that are still essential
to science and engineering students. It reflects the new excitement
in differential equations as the availability of technical computing
environments like Maple, Mathematica, and MATLAB reshape the
role and applications of the discipline. New technology has motivated
a shift in emphasis from traditional, manual methods to both qualitative
and computer-based methods that render accessible a wider range of
realistic applications. With this in mind, the text augments core
skills with conceptual perspectives that students will need for the
effective use of differential equations in their subsequent work and
NEWCoverage of seldom-used topics trimmedThere
is more streamlined coverage of certain traditional, manual topics
like exact equations and variation of parameters in Chapters 1, 3,
and 5. Symbolic, graphic, and numerical solution methods are combined
wherever it seems advantageous.
NEWContemporary topics addedElementary
introduction to period-doubling in biological and mechanical systems,
the pitchfork diagram, and the Lorenz strange attractor (all illustrated
with vivid computer graphics).
- Allows for greater emphasis on core techniques as well
as qualitative aspects of the subject associated with direction fields,
solutions curves, phase plane portraits, and dynamical systems.
NEWIncreased emphasis on and flexible
treatment of linear systems of differential equationsWith
coverage in Chapters 4 and 5 (along with the necessary linear algebra)
followed by a substantial treatment of nonlinear systems and phenomena
in Chapter 6 (including chaos in dynamical systems). Chapter 4
offers an early, intuitive introduction to first-order systems, models,
and numerical approximation techniques. Chapter 5 begins with
a self-contained treatment of the linear algebra required, then presents
the eigenvalue approach to linear systems. Section 5.5 now includes
a more extensive treatment of matrix exponentials. A new section
(5.6) on nonhomogeneous linear systems was added to this edition.
NEWAbout half of the over 300 computer-generated
graphics are newMost were constructed using MATLAB and show
vivid pictures of direction fields, solutions curves, and phase plane
portraits. For instance, the cover graphic shows an eigenfunction
of the three-dimensional wave equation that illustrates surface waves
on a spherical planet and was constructed using associated Legendre
functions (see section 10.5).
- Reflects current trends in science and engineering education
This text is unique in its blend of traditional algebraic
material with the modern geometric approachThis computer friendly
text still stresses the development of strong algebraic skills by
offering many challenging problem sets. However, the first three chapters
introduce a carefully prepared introduction to qualitative issues,
especially the geometric side.
- Brings symbolic solutions of differential equations to
Coverage begins and ends with discussions and examples
of the mathematical modeling of real-world phenomena.
Includes about 45 Computing Projects following
key sections throughout the text. Half of these are NEW or substantially
revised from the previous editionThese technology neutral
project sections contain much additional and extended problem material
and illustrate the use of computer algebra systems like Maple,
Mathematica, and MATLAB.
- Students learn through mathematical modeling and empirical
investigation to balance the questions of what equation to formulate,
how to solve it, and whether a solution will yield useful information.
The projects are expanded considerably in the Computing Projects
Manual that accompanies the text. Provided are parallel subsections
entitled Using Maple, Using Mathematica, and Using
MATLAB that detail the applicable methods and techniques of each
system. This manual is free when wrapped with text. Project notebooks
and worksheets can be downloaded from the supporting web site: www.prenhall.com/edwards
Approximately 2000 problemsover 200 of which are
NEW to this editionProblems in each section span the range
from computational problems to applied and conceptual problems. Answers
to most odd-numbered problems can be found in the answer section at
the back of the book.
- Actively engages students in the exploration and application
of computational technology. Affords them the opportunity to compare
the merits and styles of different computational systems.
Supported by a fully dedicated DE Website.
A superb direction field and phase portrait plotter is provided.
Written as Java applets, all you need is a computer with a browserno
Includes expanded versions of the in-text projects. Each project
is supported by a Maple V worksheet, a Mathematica (version 3) notebook,
and a MATLAB (version 5) script that illustrates the computing techniques
used in the project. Students can download these technology-specific
versions of the individual projects directly from the site: www.prenhall.com/edwards
An Internet Tutoravailable on Sunday nights to answer student
questions concerning the problem sets. Free to adopters.
Selected text examples are animated to provide better visual
understanding of DEs.
Quizzes, verbal in nature, are provided for each section to help
ensure that the students will actually read the core text.
1. First-Order Differential Equations.
Differential Equations and Mathematical Models. Integrals
as General and Particular Solutions. Direction Fields and Solutions
Curves. Separable Equations and Applications. Linear First-Order Equations.
Substitution Methods and Exact Equations.
2. Mathematical Models and Numerical Methods.
Population Models. Equilibrium Solutions and Stability.
Acceleration-Velocity Models. Numerical Approximation: Euler's Method.
A Closer Look at the Euler Method. The Runge-Kutta Method.
3. Linear Equations of Higher Order.
Introduction: Second-Order Linear Equations. General Solutions
of Linear Equations. Homogeneous Equations with Constant Coefficients.
Mechanical Vibrations. Nonhomogeneous Equations and the Method of
Undetermined Coefficients. Forced Oscillations and Resonance. Electrical
Circuits. Endpoint Problems and Eigenvalues.
4. Introduction to Systems of Differential Equations.
First-Order Systems and Applications. The Method of Elimination.
Numerical Methods for Systems.
5. Linear Systems of Differential Equations.
Linear Systems and Matrices. The Eigenvalue Method for Homogeneous
Systems. Second Order Systems and Mechanical Applications. Multiple
Eignvalue Solutions. Matrix Exponentials and Linear Systems. Nonhomogeneous Linear Systems.
6. Nonlinear Systems and Phenomena.
Stability and the Phase Plane. Linear and Almost Linear
Systems. Ecological Models: Predators and Competitors. Nonlinear Mechanical
Systems. Chaos in Dynamical Systems.
7. Laplace Transform Methods.
Laplace Transforms and Inverse Transforms. Transformation
of Initial Value Problems. Translation and Partial Fractions. Derivatives,
Integrals, and Products of Transforms. Periodic and Piecewise Continuous
Forcing Functions. Impulses and Delta Functions.
8. Power Series Methods.
Introduction and Review of Power Series. Series Solutions
Near Ordinary Points. Regular Singular Points. Method of Frobenius:
The Exceptional Cases. Bessel's Equation. Applications of Bessel Functions.
9. Fourier Series Methods.
Periodic Functions and Trigonometric Series. General Fourier
Series and Convergence. Fourier Sine and Cosine Series. Applications of Fourier Series. Heat Conduction and Separation of
Variables. Vibrating Strings and the One-Dimensional Wave Equation.
Steady-State Temperature and Laplace's Equation.
10. Eigenvalues and Boundary Value Problems.
Strum-Liouville Problems and Eigenfunction Expansions. Applications
of Eigenfunction Series. Steady Periodic Solutions and Natural Frequencies. Cylindrical Coordinate Problems. Higher-Dimensional Phenomena.