Fundamentals of Differential Equations presents the basic theory of differential equations and offers a variety of modern applications in science and engineering. Available in two versions, these flexible texts offer the instructor many choices in syllabus design, course emphasis (theory, methodology, applications, and numerical methods), and in using commercially available computer software. Fundamentals of Differential Equations, Eighth Edition is suitable for a one-semester sophomore- or junior-level course. Fundamentals of Differential Equations with Boundary Value Problems,'Sixth Edition, contains enough material for a two-semester course that covers and builds on boundary value problems. The Boundary Value Problems version consists of the main text plus three additional chapters (Eigenvalue Problems and Sturm-Liouville Equations; Stability of Autonomous Systems; and Existence and Uniqueness Theory).
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
