For introductory courses in Differential Equations.

This best-selling text by these well-known authors provides the conceptual development and geometric visualization of a modern differential equations course that is essential to science and engineering students. It reflects the new qualitative approach that is altering the learning of elementary differential equations, including the wide availability of scientific computing environments like Maple, Mathematica, and MATLAB. Its focus balances the traditional manual methods with the new computer-based methods that illuminate qualitative phenomena and make accessible a wider range of more realistic applications. Seldom-used topics have been trimmed and new topics added: it starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout the text.

Features

• NEW--Emphasis on new technology--Recognizes the need to instruct students in the new methods of computing differential equations.
  • Shows students the software systems tailored specifically to differential equations as well as the widely used Maple, Mathematica, and MATLAB.
• NEW--300 new computer-generated graphics--Show vivid pictures of slope fields, solution curves, and phase plane portraits.
  • Brings to life the symbolic solutions of differential equations through the visualizing of qualitative features.
• NEW--Extensive expansion of figures to the Answers Section.
  • Emphasizes the qualitative features of problem solutions.
• NEW--Application Modules--Follow key sections throughout the text; while many involve computational investigations, they are written in a technology-neutral manner. Technology-specific systems modules are available in the accompanying Applications Manual.
  • Actively engages students, providing facility and experience in the geometric visualization and qualitative interpretation that play a prominent role in contemporary differential equations.
• NEW--Fresh numerical emphasis--Made possible by the early introduction of numerical solution techniques, mathematical modeling, stability and qualitative properties of differential equations. The text includes generic numerical algorithms that can be implemented in various technologies.
  • Gives students and instructors a choice when implementing which type of technology they can utilize, as well as provides an innovative combination of topics usually dispersed later in other texts.
• NEW--Leaner and more streamlined coverage--Shaped by the availability of computational aids.
  • Allows students to learn traditional manual topics (like exact equations and variation of parameters) more easily.
• Unusually flexible treatment of linear systems--Covers in Chapters 4 and 5 the necessary linear algebra followed by a substantial treatment of nonlinear systems and phenomena in Chapter 6. The use of matrix exponential methods plays an enhanced role in this edition.
  • Reflects the current trends in science and engineering education and practice.
• Accompanying 350-page Applications Manual (Free when wrapped with text)--Provides detailed coverage of Maple, Mathematica, and MATLAB.
  • Expands and enhances for students the text's applications material.
• Approximately 2000 problems--These problems span the range from computational problems to applied and conceptual problems.
  • Provides students with problem sets that are carefully graded so that the opening problems can be easily solved by most students, giving them encouragement to continue through the set.
• DE Website--Presents projects in the form of interactive Maple, Mathematica, and MATLAB notebooks and worksheets.
  • Provides students with the opportunity to download technology-specific versions of individual projects.
• Accompanying Instructor's Solutions Manual and Student's Solutions Manual.
  • Gives both students and instructors valuable text support.

1. First Order Differential Equations.

Differential Equations and Mathematical Models. Integrals as General and Particular Solutions. Slope Fields and Solution Curves. Separable Equations and Applications. Linear First Order Equations. Substitution Methods and Exact Equations.

2. Mathematical Models and Numerical Methods.

Population Models. Equilibrium Solutions and Stability. Acceleration-Velocity Models. Numerical Approximation: Euler's Method. A Closer Look at the Euler Method, and Improvements. The Runge-Kutta Method.

3. Linear Equations of Higher Order.

Introduction: Second-Order Linear Equations. General Solutions of Linear Equations. Homogeneous Equations with Constant Coefficients. Mechanical Vibrations. Nonhomogeneous Equations and Undetermined Coefficients. Forced Oscillations and Resonance. Electrical Circuits. Endpoint Problems and Eigenvalues.

4. Introduction to Systems of Differential Equations.

First-Order Systems and Applications. The Method of Elimination. Numerical Methods for Systems.

5. Linear Systems of Differential Equations.

Linear Systems and Matrices. The Eigenvalue Method for Homogeneous Systems. Second Order Systems and Mechanical Applications. Multiple Eigenvalue Solutions. Matrix Exponentials and Linear Systems. Nonhomogenous Linear Systems.

6. Nonlinear Systems and Phenomena.

Stability and the Phase Plane. Linear and Almost Linear Systems. Ecological Models: Predators and Competitors. Nonlinear Mechanical Systems. Chaos in Dynamical Systems.

7. Laplace Transform Methods.

Laplace Transforms and Inverse Transforms. Transformation of Initial Value Problems. Translation and Partial Fractions. Derivatives, Integrals, and Products of Transforms. Periodic and Piecewise Continuous Forcing Functions. Impulses and Delta Functions.

8. Power Series Methods.

Introduction and Review of Power Series. Series Solutions Near Ordinary Points. Regular Singular Points. Method of Frobenius: The Exceptional Cases. Bessel's Equation. Applications of Bessel Functions.

9. Fourier Series Methods.

Periodic Functions and Trigonometric Series. General Fourier Series and Convergence. Even-Odd Functions and Termwise Differentiation. Applications of Fourier Series. Heat Conduction and Separation of Variables. Vibrating Strings and the One-Dimensional Wave Equation. Steady-State Temperature and Laplace's Equation.

10. Eigenvalues and Boundary Value Problems.

Sturm-Liouville Problems and Eigenfunction Expansions. Applications of Eigenfunction Series. Steady Periodic Solutions and Natural Frequencies. Applications of Bessel Functions. Higher-Dimensional Phenomena.

References.
Appendix: Existence and Uniqueness of Solutions.
Answers to Selected Problems.
Index.