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