Combining traditional differential equation material with a modern qualitative and systems approach, this new edition continues to deliver flexibility of use and extensive problem sets. The second edition's refreshed presentation includes extensive new visuals, as well as updated exercises throughout.

Features

• Unique blend of traditional algebraic and modern qualitative geometric approaches � Tempers the classic pure math approach with the practical, applied aspects of modern DE uses.
• Strong systems approach � Features a full chapter on matrix algebra, followed by three systems chapters. Practical uses of DEs today are not single equations but rather looking at several DEs using a computer.
• Novel modeling approach � Emphasizesproblems and approaches.
• Flexible use of technology � Focuses on solving and interpreting through the inclusion of several optional technology projects. The text is platform-neutral.

New To This Edition

• New Chapter (#11) on Power Series Solutions.
• Updated material throughout � Improved exposition based on current classroom trends as well as the feedback of first edition users. Includes revised coverage of exact first order equations (Ch. 2), matrix algebra (Ch. 7), and systems (Ch. 8).
• Expanded coverage of phase plane portraits � Now covered in two sections, including a new section on the trace-determinant plane.
• Large number of new figures � Help students clearly visualize ideas described in the text. The most geometric text available.
• More application-based examples � Demonstrate to students the broad applications of differential equations.
• Clearly highlighted methods of solution � Make solutions easier to identify throughout the text.
• Extensive new exercises throughout � Including all Chapters 1- 11.
• Appendix on complex numbers and matrices � Gathers information in a single location to make it more readily accessible.

Chapter 1: Introduction to Differential Equations

Differential Equation Models.
The Derivative.
Integration.

Chapter 2: First-Order Equations

Differential Equations and Solutions.
Solutions to Separable Equations.
Models of Motion.
Linear Equations.
Mixing Problems.
Exact Differential Equations.
Existence and Uniqueness of Solutions.
Dependence of Solutions on Initial Conditions.
Autonomous Equations and Stability.
Project 2.10 The Daredevil Skydiver.

Chapter 3: Modeling and Applications

Modeling Population Growth.
Models and the Real World.
Personal Finance.
Electrical Circuits.
Project 3.5 The Spruce Budworm.
Project 3.6 Social Security, Now or Later.

Chapter 4: Second-Order Equations

Definitions and Examples.
Second-Order Equations and Systems.
Linear, Homogeneous Equations with Constant Coefficients.
Harmonic Motion.
Inhomogeneous Equations; the Method of Undetermined Coefficients.
Variation of Parameters.
Forced Harmonic Motion.
Project 4.8 Nonlinear Oscillators.

Chapter 5: The Laplace Transform

The Definition of the Laplace Transform.
Basic Properties of the Laplace Transform 241.
The Inverse Laplace Transform
Using the Laplace Transform to Solve Differential Equations.
Discontinuous Forcing Terms.
The Delta Function.
Convolutions.
Summary.
Project 5.9 Forced Harmonic Oscillators.

Chapter 6: Numerical Methods

Euler's Method.
Runge-Kutta Methods.
Numerical Error Comparisons.
Practical Use of Solvers.
A Cautionary Tale.
Project 6.6 Numerical Error Comparison.

Chapter 7: Matrix Algebra

Vectors and Matrices.
Systems of Linear Equations with Two or Three Variables.
Solving Systems of Equations.
Homogeneous and Inhomogeneous Systems.
Bases of a subspace.
Square Matrices.
Determinants.

Chapter 8: An Introduction to Systems

Definitions and Examples.
Geometric Interpretation of Solutions.
Qualitative Analysis.
Linear Systems.
Properties of Linear Systems.
Project 8.6 Long-Term Behavior of Solutions.

Chapter 9: Linear Systems with Constant Coefficients

Overview of the Technique.
Planar Systems.
Phase Plane Portraits.
The Trace-Determinant Plane.
Higher Dimensional Systems.
The Exponential of a Matrix.
Qualitative Analysis of Linear Systems.
Higher-Order Linear Equations.
Inhomogeneous Linear Systems.
Project 9.10 Phase Plane Portraits.
Project 9.11 Oscillations of Linear Molecules.

Chapter 10: Nonlinear Systems

The Linearization of a Nonlinear System.
Long-Term Behavior of Solutions.
Invariant Sets and the Use of Nullclines.
Long-Term Behavior of Solutions to Planar Systems.
Conserved Quantities.
Nonlinear Mechanics.
The Method of Lyapunov.
Predator--Prey Systems.
Project 10.9 Human Immune Response to Infectious Disease.
Project 10.10 Analysis of Competing Species.

Chapter 11: Series Solutions to Differential Equations

Review of Power Series.
Series Solutions Near Ordinary Points.
Legendre's Equation.
Types of Singular Points�Euler's Equation.
Series Solutions Near Regular Singular Points.
Series Solutions Near Regular Singular Points � the General Case.
Bessel's Equation and Bessel Functions.