Fundamentals of Differential Equations with Boundary Value Problems, 7th edition

  • R Kent Nagle, 
  • Edward B. Saff, 
  • Arthur David Snider

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Fundamentals of Differential Equations and Boundary Value Problems presents the basic theory of differential equations along with modern applications in science and engineering. It adapts to various course emphases (theory, methodology, applications, numerical methods) and to use of commercially available software.

Published by Pearson (January 1st 2021) - Copyright © 2022

ISBN-13: 9780137394524

Subject: Advanced Math

Category: Differential Equations

Table of contents

1. Introduction

  • 1.1 Background
  • 1.2 Solutions and Initial Value Problems
  • 1.3 Direction Fields
  • 1.4 The Approximation Method of Euler

2. First-Order Differential Equations

  • 2.1 Introduction: Motion of a Falling Body
  • 2.2 Separable Equations
  • 2.3 Linear Equations
  • 2.4 Exact Equations
  • 2.5 Special Integrating Factors
  • 2.6 Substitutions and Transformations

3. Mathematical Models and Numerical Methods Involving First Order Equations

  • 3.1 Mathematical Modeling
  • 3.2 Compartmental Analysis
  • 3.3 Heating and Cooling of Buildings
  • 3.4 Newtonian Mechanics
  • 3.5 Electrical Circuits
  • 3.6 Improved Euler’s Method
  • 3.7 Higher-Order Numerical Methods: Taylor and Runge-Kutta

4. Linear Second-Order Equations

  • 4.1 Introduction: The Mass-Spring Oscillator
  • 4.2 Homogeneous Linear Equations: The General Solution
  • 4.3 Auxiliary Equations with Complex Roots
  • 4.4 Nonhomogeneous Equations: The Method of Undetermined Coefficients
  • 4.5 The Superposition Principle and Undetermined Coefficients Revisited
  • 4.6 Variation of Parameters
  • 4.7 Variable-Coefficient Equations
  • 4.8 Qualitative Considerations for Variable-Coefficient and Nonlinear Equations
  • 4.9 A Closer Look at Free Mechanical Vibrations
  • 4.10 A Closer Look at Forced Mechanical Vibrations

5. Introduction to Systems and Phase Plane Analysis

  • 5.1 Interconnected Fluid Tanks
  • 5.2 Elimination Method for Systems with Constant Coefficients
  • 5.3 Solving Systems and Higher-Order Equations Numerically
  • 5.4 Introduction to the Phase Plane
  • 5.5 Applications to Biomathematics: Epidemic and Tumor Growth Models
  • 5.6 Coupled Mass-Spring Systems
  • 5.7 Electrical Systems
  • 5.8 Dynamical Systems, Poincaré Maps, and Chaos

6. Theory of Higher-Order Linear Differential Equations

  • 6.1 Basic Theory of Linear Differential Equations
  • 6.2 Homogeneous Linear Equations with Constant Coefficients
  • 6.3 Undetermined Coefficients and the Annihilator Method
  • 6.4 Method of Variation of Parameters

7. Laplace Transforms

  • 7.1 Introduction: A Mixing Problem
  • 7.2 Definition of the Laplace Transform
  • 7.3 Properties of the Laplace Transform
  • 7.4 Inverse Laplace Transform
  • 7.5 Solving Initial Value Problems
  • 7.6 Transforms of Discontinuous Functions
  • 7.7 Transforms of Periodic and Power Functions
  • 7.8 Convolution
  • 7.9 Impulses and the Dirac Delta Function
  • 7.10 Solving Linear Systems with Laplace Transforms

8. Series Solutions of Differential Equations

  • 8.1 Introduction: The Taylor Polynomial Approximation
  • 8.2 Power Series and Analytic Functions
  • 8.3 Power Series Solutions to Linear Differential Equations
  • 8.4 Equations with Analytic Coefficients
  • 8.5 Cauchy-Euler (Equidimensional) Equations
  • 8.6 Method of Frobenius
  • 8.7 Finding a Second Linearly Independent Solution
  • 8.8 Special Functions

9. Matrix Methods for Linear Systems

  • 9.1 Introduction
  • 9.2 Review 1: Linear Algebraic Equations
  • 9.3 Review 2: Matrices and Vectors
  • 9.4 Linear Systems in Normal Form
  • 9.5 Homogeneous Linear Systems with Constant Coefficients
  • 9.6 Complex Eigenvalues
  • 9.7 Nonhomogeneous Linear Systems
  • 9.8 The Matrix Exponential Function

10. Partial Differential Equations

  • 10.1 Introduction: A Model for Heat Flow
  • 10.2 Method of Separation of Variables
  • 10.3 Fourier Series
  • 10.4 Fourier Cosine and Sine Series
  • 10.5 The Heat Equation
  • 10.6 The Wave Equation
  • 10.7 Laplace’s Equation

11. Eigenvalue Problems and Sturm-Liouville Equations

  • 11.1 Introduction: Heat Flow in a Non-uniform Wire
  • 11.2 Eigenvalues and Eigenfunctions
  • 11.3 Regular Sturm-Liouville Boundary Value Problems
  • 11.4 Nonhomogeneous Boundary Value Problems and the Fredholm Alternative
  • 11.5 Solution by Eigenfunction Expansion
  • 11.6 Green’s Functions
  • 11.7 Singular Sturm-Liouville Boundary Value Problems.
  • 11.8 Oscillation and Comparison Theory

12. Stability of Autonomous Systems

  • 12.1 Introduction: Competing Species
  • 12.2 Linear Systems in the Plane
  • 12.3 Almost Linear Systems
  • 12.4 Energy Methods
  • 12.5 Lyapunov’s Direct Method
  • 12.6 Limit Cycles and Periodic Solutions
  • 12.7 Stability of Higher-Dimensional Systems

13. Existence and Uniqueness Theory

  • 13.1 Introduction: Successive Approximations
  • 13.2 Picard’s Existence and Uniqueness Theorem
  • 13.3 Existence of Solutions of Linear Equations
  • 13.4 Continuous Dependence of Solutions

Appendix A Review of Integration Techniques

Appendix B Newton’s Method

Appendix C Simpson’s Rule

Appendix D Cramer’s Rule

Appendix E Method of Least Squares

Appendix F Runge-Kutta Procedure for n Equations

Appendix G Software for Analyzing Differential Equations

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