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