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Differential Equations

Theory, Technique and Practice, Second Edition

## Preview

## 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 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.

## Table of Contents

**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

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

Second-Order Linear Equations

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

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 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 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

Sturm-Liouville 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 Impulsively-Started 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 Runge-Kutta Method

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

Problems for Review and Discovery

Systems of First-Order Equations

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

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 Poincare-Bendixson 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 Poincare-Bendixson

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 well-known 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 MATH1316"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