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Differential equations : theory, technique, and practice

By: Krantz, Steven G.
Material type: materialTypeLabelBookSeries: Textbooks in Mathematics. Publisher: Boca Raton CRC Press 2015Edition: 2nd ed.Description: xvi, 541 p.ISBN: 9781482247022.Subject(s): Differential equations | MathematicsDDC classification: 515.35 Summary: 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. (https://www.crcpress.com/Differential-Equations-Theory-Technique-and-Practice-Second-Edition/Krantz/9781482247022)
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Table of Contents


1. 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


2. 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


3. 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



4. POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS



• Introduction and Review of Power Series

o 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


5. Numerical Methods


• Introductory Remarks

• The Method of Euler

• The Error Term

• An Improved Euler Method

• The Runge-Kutta Method

• Anatomy of an Application:

• Problems for Review and Discovery


6. 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


7. 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


8. 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



9. 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


10. 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


11. 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


12. 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



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.

(https://www.crcpress.com/Differential-Equations-Theory-Technique-and-Practice-Second-Edition/Krantz/9781482247022)

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