Differential Equations
Theory, Technique and Practice, Second Edition
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Book Description
"Krantz is a very prolific writer. He … creates excellent examples and problem sets."
—Albert Boggess, Professor and Director of the School of Mathematics and Statistical Sciences, Arizona State University, Tempe, USA
Designed for a one or twosemester undergraduate course, Differential Equations: Theory, Technique and Practice, Second Edition educates a new generation of mathematical scientists and engineers on differential equations. This edition continues to emphasize examples and mathematical modeling as well as promote analytical thinking to help students in future studies.
New to the Second Edition
 Improved exercise sets and examples
 Reorganized material on numerical techniques
 Enriched presentation of predatorprey problems
 Updated material on nonlinear differential equations and dynamical systems
 A new appendix that reviews linear algebra
In each chapter, lively historical notes and mathematical nuggets enhance students’ reading experience by offering perspectives on the lives of significant contributors to the discipline. "Anatomy of an Application" sections highlight rich applications from engineering, physics, and applied science. Problems for review and discovery also give students some openended material for exploration and further learning.
Table of Contents
Preface
What is a Differential Equation?
Introductory Remarks
The Nature of Solutions
Separable Equations
FirstOrder Linear Equations
Exact Equations
Orthogonal Trajectories and Families of Curves
Homogeneous Equations
Integrating Factors
Reduction of Order
Dependent Variable Missing
Independent Variable Missing
The Hanging Chain and Pursuit Curves
The Hanging Chain
Pursuit Curves
Electrical Circuits
Anatomy of an Application: The Design of a Dialysis Machine
Problems for Review and Discovery
SecondOrder Linear Equations
SecondOrder Linear Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
The Use of a Known Solution to Find Another
Vibrations and Oscillations
Undamped Simple Harmonic Motion
Damped Vibrations
Forced Vibrations
A Few Remarks about Electricity
Newton’s Law of Gravitation and Kepler’s Laws
Kepler’s Second Law
Kepler’s First Law
Kepler’s Third Law
Higher Order Equations
Historical Note: Euler
Anatomy of an Application: Bessel Functions and the Vibrating Membrane
Problems for Review and Discovery
Qualitative Properties and Theoretical Aspects
A Bit of Theory
Picard’s Existence and Uniqueness Theorem
The Form of a Differential Equation
Picard’s Iteration Technique
Some Illustrative Examples
Estimation of the Picard Iterates
Oscillations and the Sturm Separation Theorem
The Sturm Comparison Theorem
Anatomy of an Application: The Green’s Function
Problems for Review and Discovery
Power Series Solutions and Special Functions
Introduction and Review of Power Series
Review of Power Series
Series Solutions of FirstOrder Equations
SecondOrder Linear Equations: Ordinary Points
Regular Singular Points
More on Regular Singular Points
Gauss’s Hypergeometric Equation
Historical Note: Gauss
Historical Note: Abel
Anatomy of an Application: Steady State Temperature in a Ball
Problems for Review and Discovery
Fourier Series: Basic Concepts
Fourier Coefficients
Some Remarks about Convergence
Even and Odd Functions: Cosine and Sine Series
Fourier Series on Arbitrary Intervals
Orthogonal Functions
Historical Note: Riemann
Anatomy of an Application: Introduction to the Fourier Transform
Problems for Review and Discovery
Partial Differential Equations and Boundary Value Problems
Introduction and Historical Remarks
Eigenvalues, Eigenfunctions, and the Vibrating String
Boundary Value Problems
Derivation of the Wave Equation
Solution of the Wave Equation
The Heat Equation
The Dirichlet Problem for a Disc
The Poisson Integral
SturmLiouville Problems
Historical Note: Fourier
Historical Note: Dirichlet
Anatomy of an Application: Some Ideas from Quantum Mechanics
Problems for Review and Discovery
Laplace Transforms
Introduction
Applications to Differential Equations
Derivatives and Integrals of Laplace Transforms
Convolutions
Abel’s Mechanics Problem
The Unit Step and Impulse Functions
Historical Note: Laplace
Anatomy of an Application: Flow Initiated by an ImpulsivelyStarted Flat Plate
Problems for Review and Discovery
The Calculus of Variations
Introductory Remarks
Euler’s Equation
Isoperimetric Problems and the Like
Lagrange Multipliers
Integral Side Conditions
Finite Side Conditions
Historical Note: Newton
Anatomy of an Application: Hamilton’s Principle and its Implications
Problems for Review and Discovery
Numerical Methods
Introductory Remarks
The Method of Euler
The Error Term
An Improved Euler Method
The RungeKutta Method
Anatomy of an Application: A Constant Perturbation Method for Linear, SecondOrder Equations
Problems for Review and Discovery
Systems of FirstOrder Equations
Introductory Remarks
Linear Systems
Homogeneous Linear Systems with Constant Coefficients
Nonlinear Systems: Volterra’s PredatorPrey Equations
Solving HigherOrder Systems Using Matrix Theory
Anatomy of an Application: Solution of Systems with Matrices and Exponentials
Problems for Review and Discovery
The Nonlinear Theory
Some Motivating Examples
Specializing Down
Types of Critical Points: Stability
Critical Points and Stability for Linear Systems
Stability by Liapunov’s Direct Method
Simple Critical Points of Nonlinear Systems
Nonlinear Mechanics: Conservative Systems
Periodic Solutions: The PoincareBendixson Theorem
Historical Note: Poincare
Anatomy of an Application: Mechanical Analysis of a Block on a Spring
Problems for Review and Discovery
Dynamical Systems
Flows
Dynamical Systems
Stable and Unstable Fixed Points
Linear Dynamics in the Plane
Some Ideas from Topology
Open and Closed Sets
The Idea of Connectedness
Closed Curves in the Plane
Planar Autonomous Systems
Ingredients of the Proof of PoincareBendixson
Anatomy of an Application: Lagrange’s Equations
Problems for Review and Discovery
Appendix on Linear Algebra
Vector Spaces
The Concept of Linear Independence
Bases
Inner Product Spaces
Linear Transformations and Matrices
Eigenvalues and Eigenvectors
Bibliography
Author(s)
Biography
Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has written more than 65 books and more than 175 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.
Reviews
"Retaining many of the strong aspects of the first edition, which received positive feedback from readers, the new edition focuses on clarity of exposition and examples, many of which feature applications of differential equations. … Being an homage to the excellent writing skills of George Simmons and his wellknown text on differential equations written back in 1972, this updated edition maintains the highest standards of mathematics exposition. Warmly recommended as a comprehensive and modern textbook on theory, methods, and applications of differential equations!"
—Zentralblatt MATH 1316"Krantz is a very prolific writer. He…creates excellent examples and problem sets."
—Albert Boggess, Professor and Director of the School of Mathematics and Statistical Sciences, Arizona State University, Tempe, USA
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