Fundamentals of Differential Equations and Boundary Value Problems

1. Introduction

1.1 Background

1.2 Solutions and Initial Value Problems

1.3 Direction Fields

1.4 The Approximation Method of Euler

           Chapter Summary

           Technical Writing Exercises

           Group Projects for Chapter 1

           A. Taylor Series Method

           B. Picard's Method

           C. The Phase Line

 

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

           Chapter Summary

           Review Problems

           Technical Writing Exercises

           Group Projects for Chapter 2

           A. Oil Spill in a Canal

           B. Differential Equations in Clinical Medicine

           C. Torricelli's Law of Fluid Flow

           D. The Snowplow Problem

           E. Two Snowplows

           F. Clairaut Equations and Singular Solutions

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

           H. Utility Functions and Risk Aversion

           I. Designing a Solar Collector

           J. Asymptotic Behavior of Solutions to Linear Equations

 

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

           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

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

           Chapter Summary

           Review Problems

           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. Asymptotic Behavior of Solutions

 

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

           Chapter Summary

           Review Problems

           Group Projects for Chapter 5

           A. Designing a Landing System for Interplanetary Travel

           B. Spread of Staph Infections in Hospitals-Part 1

           C. Things That Bob

           D. Hamiltonian Systems

           E. Cleaning Up the Great Lakes

 

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

           Chapter Summary

           Review Problems

           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

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 and Periodic Functions

7.7 Convolution

7.8 Impulses and the Dirac Delta Function

7.9 Solving Linear Systems with Laplace Transforms

           Chapter Summary

           Review Problems

           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

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

           Chapter Summary

           Review Problems

           Technical Writing Exercises

           Group Projects for Chapter 8

           A. Alphabetization Algorithms

           B. Spherically Symmetric Solutions to Shrödinger's Equation for the Hydrogen Atom

           C. Airy's Equation

           D. Buckling of a Tower

           E. Aging Spring and Bessel 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

           Chapter Summary

           Review Problems

           Technical Writing Exercises

           Group Projects for Chapter 9

           A. Uncoupling Normal Systems

           B. Matrix Laplace Transform Method

           C. Undamped Second-Order Systems

           D. Undetermined Coefficients for System Forced by Homogeneous

 

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

           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

11.1 Introduction: Heat Flow in a Nonuniform 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

           Chapter Summary

           Review Problems

           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

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

Chapter Summary

Review Problems

           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

           E. Spread of Staph Infections in Hospitals-Part 2

           F. A Growth Model for Phytoplankton-Part 2

 

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

           Chapter Summary

           Review Problems

           Technical Writing Exercises

 

Appendices

A. Review of Integration Techniques

B. Newton's Method

C. Simpson's Rule

D. Cramer's Rule

E. Method of Least Squares

F. Runge-Kutta Procedure for n Equations

 

Answers to Odd-Numbered Problems

Index