Differential Equations
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Differential Equations : Theory, Technique and Practice, Second Edition

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"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.
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Product details

  • Hardback | 557 pages
  • 156 x 235 x 30.48mm | 930g
  • Oakville, Canada
  • English
  • New edition
  • 2nd New edition
  • 9 Tables, black and white; 156 Illustrations, black and white
  • 148224702X
  • 9781482247022
  • 2,071,204

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
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Review quote

"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 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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About Steven G. Krantz

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.
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