A Course in Ordinary Differential Equations

ISBN-13: 9781466509085 / Angielski / Twarda / 2014 / 807 str.

Randall J. Swift; Stephen A. Wirkus

Opis

A Course in Ordinary Differential Equations, Second Edition teaches students how to use analytical and numerical solution methods in typical engineering, physics, and mathematics applications. Lauded for its extensive computer code and student-friendly approach, the first edition of this popular textbook was the first on ordinary differential equations (ODEs) to include instructions on using MATLAB^(R), Mathematica^(R), and Maple(TM). This second edition reflects the feedback of students and professors who used the first edition in the classroom. New to the Second Edition

• Moves the computer codes to Computer Labs at the end of each chapter, which gives professors flexibility in using the technology
• Covers linear systems in their entirety before addressing applications to nonlinear systems
• Incorporates the latest versions of MATLAB, Maple, and Mathematica
• Includes new sections on complex variables, the exponential response formula for solving nonhomogeneous equations, forced vibrations, and nondimensionalization
• Highlights new applications and modeling in many fields
• Presents exercise sets that progress in difficulty
• Contains color graphs to help students better understand crucial concepts in ODEs
• Provides updated and expanded projects in each chapter

Suitable for a first undergraduate course, the book includes all the basics necessary to prepare students for their future studies in mathematics, engineering, and the sciences. It presents the syntax from MATLAB, Maple, and Mathematica to give students a better grasp of the theory and gain more insight into real-world problems. Along with covering traditional topics, the text describes a number of modern topics, such as direction fields, phase lines, the Runge-Kutta method, and epidemiological and ecological models. It also explains concepts from linear algebra so that students acquire a thorough understanding of differential equations.

Spis treści

Traditional First-Order Differential Equations
Introduction to First-Order Equations
Separable Differential Equations
Linear Equations
Some Physical Models Arising as Separable Equations
Exact Equations
Special Integrating Factors and Substitution Methods

Geometrical and Numerical Methods for First-Order Equations
Direction Fields—the Geometry of Differential Equations
Existence and Uniqueness for First-Order Equations
First-Order Autonomous Equations—Geometrical Insight
Modeling in Population Biology
Numerical Approximation: Euler and Runge-Kutta Methods
An Introduction to Autonomous Second-Order Equations

Elements of Higher-Order Linear Equations
Introduction to Higher-Order Equations
Linear Independence and the Wronskian
Reduction of Order—the Case n = 2
Numerical Considerations for nth-Order Equations
Essential Topics from Complex Variables
Homogeneous Equations with Constant Coefficients
Mechanical and Electrical Vibrations

Techniques of Nonhomogeneous Higher-Order Linear Equations
Nonhomogeneous Equations
Method of Undetermined Coefficients via Superposition
Method of Undetermined Coefficients via Annihilation
Exponential Response and Complex Replacement
Variation of Parameters
Cauchy-Euler (Equidimensional) Equation
Forced Vibrations

Fundamentals of Systems of Differential Equations
Useful Terminology
Gaussian Elimination
Vector Spaces and Subspaces
Eigenvalues and Eigenvectors
A General Method, Part I: Solving Systems with Real & Distinct or Complex Eigenvalues
A General Method, Part II: Solving Systems with Repeated Real Eigenvalues
Matrix Exponentials
Solving Linear Nonhomogeneous Systems of Equations

Geometrical and Numerical Methods for First-Order Equations
An Introduction to the Phase Plane
Nonlinear Equations and Phase Plane Analysis
Bifurcations
Epidemiological Models
Models in Ecology

Laplace Transforms
Introduction
Fundamentals of the Laplace Transform
The Inverse Laplace Transform
Translated Functions, Delta Function, and Periodic Functions
The s-Domain and Poles
Solving Linear Systems using Laplace Transforms
The Convolution

Series Methods
Power Series Representations of Functions
The Power Series Method
Ordinary and Singular Points
The Method of Frobenius
Bessel Functions

Appendix A: An Introduction to MATLAB, Maple, and Mathematica
Appendix B: Selected Topics from Linear Algebra

Answers to Odd Problems

References

Index

A Review, Computer Labs, and Projects appear at the end of each chapter.