Differential Equations (2^nd Ed.)
Theory, Technique and Practice, Second Edition
Textbooks in Mathematics Series

"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 two-semester 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 predator-prey 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 open-ended material for exploration and further learning.

Preface

What is a Differential Equation?

Introductory Remarks

The Nature of Solutions

Separable Equations

First-Order Linear Equations

Exact Equations

Orthogonal Trajectories and Families of Curves

Homogeneous Equations

Integrating Factors

Reduction of Order

The Hanging Chain and Pursuit Curves

Electrical Circuits

Anatomy of an Application: The Design of a Dialysis Machine

Problems for Review and Discovery

Second-Order 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

Newton’s Law of Gravitation and Kepler’s Laws

Higher Order Equations

Historical Note: Euler

Anatomy of an Application: Bessel Functions and the Vibrating Membrane

Problems for Review and Discovery

A Bit of Theory

Picard’s Existence and Uniqueness Theorem

Oscillations and the Sturm Separation Theorem

The Sturm Comparison Theorem

Anatomy of an Application: The Green’s Function

Problems for Review and Discovery

Introduction and Review of Power Series

Series Solutions of First-Order Equations

Second-Order 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 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

Introduction and Historical Remarks

Eigenvalues, Eigenfunctions, and the Vibrating String

The Heat Equation

The Dirichlet Problem for a Disc

Sturm-Liouville Problems

Historical Note: Fourier

Historical Note: Dirichlet

Anatomy of an Application: Some Ideas from Quantum Mechanics

Problems for Review and Discovery

Introduction

Applications to Differential Equations

Derivatives and Integrals of Laplace Transforms

Convolutions

The Unit Step and Impulse Functions

Historical Note: Laplace

Anatomy of an Application: Flow Initiated by an Impulsively-Started Flat Plate

Problems for Review and Discovery

Introductory Remarks

Euler’s Equation

Isoperimetric Problems and the Like

Historical Note: Newton

Anatomy of an Application: Hamilton’s Principle and its Implications

Problems for Review and Discovery

Introductory Remarks

The Method of Euler

The Error Term

An Improved Euler Method

The Runge-Kutta Method

Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equations

Problems for Review and Discovery

Introductory Remarks

Linear Systems

Homogeneous Linear Systems with Constant Coefficients

Nonlinear Systems: Volterra’s Predator-Prey Equations

Solving Higher-Order Systems Using Matrix Theory

Anatomy of an Application: Solution of Systems with Matrices and Exponentials

Problems for Review and Discovery

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 Poincare-Bendixson Theorem

Historical Note: Poincare

Anatomy of an Application: Mechanical Analysis of a Block on a Spring

Problems for Review and Discovery

Flows

Some Ideas from Topology

Planar Autonomous Systems

Anatomy of an Application: Lagrange’s Equations

Problems for Review and Discovery

Vector Spaces

The Concept of Linear Independence

Bases

Inner Product Spaces

Linear Transformations and Matrices

Eigenvalues and Eigenvectors

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.