Riassunto
The first contemporary textbook on ordinary differential equations (ODEs) to include instructions on MATLAB®, Mathematica®, and Maple™, A Course in Ordinary Differential Equations focuses on applications and methods of analytical and numerical solutions, emphasizing approaches used in the typical engineering, physics, or mathematics student's field of study. Stressing applications wherever possible, the authors have written this text with the applied math, engineer, or science major in mind. It includes a number of modern topics that are not commonly found in a traditional sophomore-level text. For example, Chapter 2 covers direction fields, phase line techniques, and the Runge-Kutta method; another chapter discusses linear algebraic topics, such as transformations and eigenvalues. Chapter 6 considers linear and nonlinear systems of equations from a dynamical systems viewpoint and uses the linear algebra insights from the previous chapter; it also includes modern applications like epidemiological models. With sufficient problems at the end of each chapter, even the pure math major will be fully challenged. Although traditional in its coverage of basic topics of ODEs, A Course in Ordinary Differential Equations is one of the first texts to provide relevant computer code and instruction in MATLAB, Mathematica, and Maple that will prepare students for further study in their fields.
Contenuti
TRADITIONAL FIRST ORDER DIFFERENTIAL EQUATIONS
Some Basic Terminology
Separable Differential Equations
Some Physical Problems arising as Separable Equations
Exact Equations
Linear Equations
GEOMETRICAL & 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
Population Modeling: An Application of Autonomous Equations
Numerical Approximation with the Euler Method
Numerical Approximation with the Runge-Kutta Method
An Introduction to Autonomous Second Order Equations
ELEMENTS OF HIGHER ORDER LINEAR EQUATIONS
Some Terminology
Essential Topics from Linear Algebra
Reduction of Order - The Case n=2
Operator Notation
Numerical Considerations for nth Order Equations
TECHNIQUES OF HIGHER ORDER LINEAR EQUATIONS
Homogeneous Equations with Constant Coefficients
A Mass on a Spring
Cauchy-Euler (Equidimensional) Equation
Nonhomogeneous Equations
The Method of Undetermined Coefficients via Tables
The Method of Undetermined Coefficients via the Annihilator Method
Variation of Parameters
FUNDAMENTALS OF SYSTEMS OF DIFFERENTIAL EQUATIONS
Systems of Two Equations - Motivational Examples
Useful Terminology
Linear Transformations and the Fundamental Subspaces
Eigenvalues and Eigenvectors
Matrix Exponentials
TECHNIQUES OF SYSTEMS OF DIFFERENTIAL EQUATIONS
A General Method, Part I: Solving Systems with Real, Distinct Eigenvalues
A General Method, Part II: Solving Systems with Repeated Real or Complex Eigenvalues
Solving Linear Homogeneous and Nonhomogeneous Systems of Equations
Nonlinear Equations and Phase Plane Analysis
Epidemiological Models
LAPLACE TRANSFORMS
Fundamentals of the Laplace Transform
Properties of the Laplace Transforms
Step Functions, Translated Functions, and Periodic Functions
The Inverse Laplace Transform
Laplace Transform Solution of Linear Differential Equations
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: Graphing Factored Polynomials
Appendix C: Selected Topics from Linear Algebra
Appendix D: Answers to Selected Exercises
All chapters have additional problems and each chapter has its own project(s).
Le informazioni nella sezione "Su questo libro" possono far riferimento a edizioni diverse di questo titolo.
