Fundamentals of Differential Equations with Boundary Value Problems

1. Introduction

Background

Solutions and Initial Value Problems

Direction Fields

The Approximation Method of Euler

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 1

            A. Taylor Series Method

            B. Picard's Method

            C. Magnetic Dipole

            D. The Phase Line

 

2. First-Order Differential Equations

Introduction: Motion of a Falling Body

Separable Equations

Linear Equations

Exact Equations

Special Integrating Factors

Substitutions and Transformations

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 2

            A. Differential Equations in Clinical Medicine

            B. Torricelli's Law of Fluid Flow

            C. The Snowplow Problem

            D. Two Snowplows

            E. Clairaut Equations and Singular Solutions

            F. Multiple Solutions of a First-Order Initial Value Problem

            G. Designing a Solar Collector

            H. Asymptotic Behavior of Solutions to Linear Equations

            I. Utility Functions and Risk Aversion

 

3. Mathematical Models and Numerical Methods Involving First Order Equations

Mathematical Modeling

Compartmental Analysis

Heating and Cooling of Buildings

Newtonian Mechanics

Electrical Circuits

Improved Euler's Method

Higher-Order Numerical Methods: Taylor and Runge-Kutta

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 3

            A. Dynamics of HIV Infection

            B. Aquaculture

            C. Curve of Pursuit

            D. Aircraft Guidance in a Crosswind

            E. Feedback and the Op Amp

            F. Bang-Bang Controls

            G. Market Equilibrium: Stability and Time Paths

            H. Stability of Numerical Methods

            I. Period Doubling and Chaos

 

4. Linear Second-Order Equations

Introduction: The Mass-Spring Oscillator

Homogeneous Linear Equations; The General Solution

Auxiliary Equations with Complex Roots

Nonhomogeneous Equations: The Method of Undetermined Coefficients

The Superposition Principle and Undetermined Coefficients Revisited

Variation of Parameters

Variable-Coefficient Equations

Qualitative Considerations for Variable-Coefficient and Nonlinear Equations

A Closer Look at Free Mechanical Vibrations

A Closer Look at Forced Mechanical Vibrations

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 4

            A. Nonlinear Equations Solvable by First-Order Techniques

            B. Apollo Reentry

            C. Simple Pendulum

            D. Linearization of Nonlinear Problems

            E. Convolution Method

            F. Undetermined Coefficients Using Complex Arithmetic

            G. An Alternative to the Method of Undetermined Coefficients

            H. Asymptotic Behavior of Solutions

 

 

5. Introduction to Systems and Phase Plane Analysis

Interconnected Fluid Tanks

Elimination Method for Systems with Constant Coefficients

Solving Systems and Higher-Order Equations Numerically

Introduction to the Phase Plane

Applications to Biomathematics: Epidemic and Tumor Growth Models

Coupled Mass-Spring Systems

Electrical Systems

Dynamical Systems, Poincaré Maps, and Chaos

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 5

            A. Designing a Landing System for Interplanetary Travel

            B. Things That Bob

            C. Hamiltonian Systems

            D. Strange Behavior of Competing Species - Part 1

            E. Cleaning Up the Great Lakes

 

6. Theory of Higher-Order Linear Differential Equations

Basic Theory of Linear Differential Equations

Homogeneous Linear Equations with Constant Coefficients

Undetermined Coefficients and the Annihilator Method

Method of Variation of Parameters

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 6

            A. Computer Algebra Systems and Exponential Shift

            B. Justifying the Method of Undetermined Coefficients

            C. Transverse Vibrations of a Beam

 

7. Laplace Transforms

Introduction: A Mixing Problem

Definition of the Laplace Transform

Properties of the Laplace Transform

Inverse Laplace Transform

Solving Initial Value Problems

Transforms of Discontinuous and Periodic Functions

Convolution

Impulses and the Dirac Delta Function

Solving Linear Systems with Laplace Transforms

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 7

            A. Duhamel's Formulas

            B. Frequency Response Modeling

            C. Determining System Parameters

 

8. Series Solutions of Differential Equations

Introduction: The Taylor Polynomial Approximation

Power Series and Analytic Functions

Power Series Solutions to Linear Differential Equations

Equations with Analytic Coefficients

Cauchy-Euler (Equidimensional) Equations

Method of Frobenius

Finding a Second Linearly Independent Solution

Special Functions

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 8

            A. Spherically Symmetric Solutions to Shrodinger's Equation for the Hydrogen Atom

            B. Airy's Equation

            C. Buckling of a Tower

            D. Aging Spring and Bessel Functions

 

9. Matrix Methods for Linear Systems

Introduction

Review 1: Linear Algebraic Equations

Review 2: Matrices and Vectors

Linear Systems in Normal Form

Homogeneous Linear Systems with Constant Coefficients

Complex Eigenvalues

Nonhomogeneous Linear Systems

The Matrix Exponential Function

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 9

            A. Uncoupling Normal Systems

            B. Matrix Laplace Transform Method

            C. Undamped Second-Order Systems

            D. Strange Behavior of Competing Species - Part II

 

10. Partial Differential Equations

Introduction: A Model for Heat Flow

Method of Separation of Variables

Fourier Series

Fourier Cosine and Sine Series

The Heat Equation

The Wave Equation

Laplace's Equation

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 10

            A. Steady-State Temperature Distribution in a Circular Cylinder

            B. A Laplace Transform Solution of the Wave Equation

            C. Green's Function

            D. Numerical Method for ?u=f on a Rectangle

 

11. Eigenvalue Problems and Sturm-Liouville Equations

Introduction: Heat Flow in a Nonuniform Wire

Eigenvalues and Eigenfunctions

Regular Sturm-Liouville Boundary Value Problems

Nonhomogeneous Boundary Value Problems and the Fredholm Alternative

Solution by Eigenfunction Expansion

Green's Functions.

Singular Sturm-Liouville Boundary Value Problems.

Oscillation and Comparison Theory.

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 11

            A. Hermite Polynomials and the Harmonic Oscillator

            B. Continuous and Mixed Spectra

            C. Picone Comparison Theorem

            D. Shooting Method

            E. Finite-Difference Method for Boundary Value Problems

 

12. Stability of Autonomous Systems

Introduction: Competing Species

Linear Systems in the Plane

Almost Linear Systems

Energy Methods

Lyapunov's Direct Method

Limit Cycles and Periodic Solutions

Stability of Higher-Dimensional Systems

Chapter Summary

Technical Writing Exercises

Group Projects for Chapter 12

            A. Solutions and Korteweg-de Vries Equation

            B. Burger's Equation

            C. Computing Phase Plane Diagrams

            D. Ecosystem on Planet GLIA-2

 

13. Existence and Uniqueness Theory

Introduction: Successive Approximations

Picard's Existence and Uniqueness Theorem

Existence of Solutions of Linear Equations

Continuous Dependence of Solutions

Chapter Summary

Technical Writing Exercises

 

Appendices

A. Newton's Method

B. Simpson's Rule

C. Cramer's Rule

D. Method of Least Squares

E. Runge-Kutta Precedure for n Equations

 

Answers to Odd-Numbered Problems

Index