Incorporating a modeling approach throughout, this exciting text emphasizes concepts and shows that the study
of differential equations is a beautiful application of the ideas and techniques of calculus to everyday life.
By taking advantage of readily available technology, the authors eliminate most of the specialized techniques for
deriving formulas for solutions found in traditional texts and replace them with topics that focus on the formulation
of differential equations and the interpretations of their solutions. Students will generally attack a given equation
from three different points of view to obtain an understanding of the solutions: qualitative, numeric, and analytic.
Since many of the most important differential equations are nonlinear, students learn that numerical and qualitative
techniques are more effective than analytic techniques in this setting. Overall, students discover how to identify
and work effectively with the mathematics in everyday life, and they learn how to express the fundamental principles
that govern many phenomena in the language of differential equations.
Table of Contents
1. FIRST-ORDER DIFFERENTIAL EQUATIONS.
Modeling via Differential Equations. Analytic Technique: Separation of Variables. Qualitative Technique: Slope
Fields. Numerical Technique: Euler's Method. Existence and Uniqueness of Solutions. Equilibria and the Phase Line.
Bifurcations. Linear Equations. Integration Factors for Linear Equations. Review Exercises for Chapter 1. Labs
for Chapter 1.
2. FIRST-ORDER SYSTEMS.
Modeling via Systems. The Geometry of Systems. Analytic Methods for Special Systems. Euler's Method for Systems.
The Lorenz Equations. Review Exercises for Chapter 2. Labs for Chapter 2.
3. LINEAR SYSTEMS.
Properties of Linear Systems and the Linearity Principle. Straight-Line Solutions. Phase Planes for Linear Systems
with Real Eigenvalues. Complex Eigenvalues. Special Cases: Repeated and Zero Eigenvalues. Second-Order Linear Equations.
The Trace-Determinant Plane. Linear Systems in Three Dimensions. Review Exercises for Chapter 3. Labs for Chapter
3.
4. FORCING AND RESONANCE.
Forced Harmonic Oscillators. Sinusoidal Forcing. Undamped Forcing and Resonance. Amplitude and Phase of the
Steady State. The Tacoma Narrows Bridge. Review Exercises for Chapter 4. Labs for Chapter 4.
5. NONLINEAR SYSTEMS.
Equilibrium Point Analysis. Qualitative Analysis. Hamiltonian Systems. Dissipative Systems. Nonlinear Systems
in Three Dimensions. Periodic Forcing of Nonlinear Systems and Chaos. Review Exercises for Chapter 5. Labs for
Chapter 5.
6. LAPLACE TRANSFORMS.
Laplace Transforms. Discontinuous Functions. Second-Order Equations. Delta Functions and Impulse Forcing. Convolutions.
The Qualitative Theory of Laplace Transforms. Review Exercises for Chapter 6. Labs for Chapter 6.
7. NUMERICAL METHODS.
Numerical Error in Euler's Method. Improving Euler's Method. The Runge-Kutta Method. The Effects of Finite Arithmetic.
Review Exercises for Chapter 7. Labs for Chapter 7.
8. DISCRETE DYNAMICAL SYSTEMS.
The Discrete Logistic Equation. Fixed Points and Periodic Points. Bifurcations. Chaos. Chaos in the Lorenz System.
Review Exercises for Chapter 8. Labs for Chapter 8. Hints and Answers.
