Differential Equations with Linear Algebra

ISBN-10: 0195385861
ISBN-13: 9780195385861
Edition: 2009
List price: $100.00
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Description: Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we  More...

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Book details

List price: $100.00
Copyright year: 2009
Publisher: Oxford University Press, Incorporated
Publication date: 11/5/2009
Binding: Hardcover
Pages: 576
Size: 6.25" wide x 9.25" long x 1.25" tall
Weight: 2.508
Language: English

Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomorelevel, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, the book is appropriate for courses for majors in mathematics, science, and engineering that study systems of differential equations. Because of its emphasis on linearity, the text opens with a full chapter devoted to essential ideas in linear algebra. Motivated by future problems in systems of differential equations, the chapter on linear algebra introduces such key ideas as systems of algebraic equations, linear combinations, the eigenvalue problem, and bases and dimension of vector spaces. This chapter enables students to quickly learn enough linear algebra to appreciate the structure of solutions to linear differentialequations and systems thereof in subsequent study and to apply these ideas regularly. The book offers an example-driven approach, beginning each chapter with one or two motivating problems that are applied in nature. The following chapter develops the mathematics necessary to solve these problems and explores related topics further. Even in more theoretical developments, we use an example-first style to build intuition and understanding before stating or proving general results. Over 100 figures provide visual demonstration of key ideas; the use of the computer algebrasystem Maple and Microsoft Excel are presented in detail throughout to provide further perspective and support students' use of technology in solving problems. Each chapter closes with several substantial projects for further study, many of which are based in applications.

Merle C. Potter holds a B.S. in Mechanical Engineering and an M.S. in Engineering Mechanics from Michigan Technological University, an M.S. in Aerospace Engineering and a PhD in Engineering Mechanics from the University of Michigan. Dr. Potter taught for 40 years, 33 of those years spent at Michigan State University, which he joined in 1965. He teaches thermodynamics, fluid mechanics and numerous other courses. He has authored and co-authored 35 textbooks, help books, and engineering exam review books. He has performed research in fluid flow stability and energy. Dr. Potter has received numerous awards, including the Ford Faculty Scholarship, Teacher-Scholar Award, ASME Centennial Award . and the MSU Mechanical Engineering Faculty Award. He is a member of Tau Beta Pi, Phi Eta Sigma, Phi Kappa Phi, Pi Tau Sigma, Sigma Xi, the ASEE, ASME, and American Academy of Mechanics.

Introduction
Essentials of linear algebra
Motivating problems
Systems of linear equations
Row reduction using Maple
Linear combinations
Markov chains: an application of matrix-vector multiplication
Matrix products using Maple
The span of a set of vectors
Systems of linear equations revisited
Linear independence
Matrix algebra
Matrix algebra using Maple
The inverse of a matrix
Computer graphics
Matrix inverses using Maple
The determinant of a matrix
Determinants using Maple
The eigenvalue problem
Markov chains, eigenvectors, and Google
Using Maple to find eigenvalues and eigenvectors
Generalized vectors
Bases and dimension in vector spaces
For further study
Computer graphics: geometry and linear algebra at work
B�zier curves
Discrete dynamical systems
First-order differential equations
Motivating problems
Definitions, notation, and terminology
Plotting slope fields using Maple
Linear first-order differential equations
Applications of linear first-order differential equations
Mixing problems
Exponential growth and decay
Newton's law of Cooling
Nonlinear first-order differential equations
Separable equations
Exact equations
Euler's method
Implementing Euler's method in Excel
Applications of nonlinear first-order differential equations
The logistic equation
Torricelli's law
For further study
Converting certain second-order des to first-order DEs
How raindrops fall
Riccati's equation
Bernoulli's equation
Linear systems of differential equations
Motivating problems
The eigenvalue problem revisited
Homogeneous linear first-order systems
Systems with all real linearly independent eigenvectors
Plotting direction fields for systems using Maple
When a matrix lacks two real linearly independent eigenvectors
Nonhomogeneous systems: undetermined coefficients
Nonhomogeneous systems: variation of parameters
Applying variation of parameters using Maple
Applications of linear systems
Mixing problems
Spring-mass systems
RLC circuits
For further study
Diagonalizable matrices and coupled systems
Matrix exponential
Higher order differential equations
Motivating equations
Homogeneous equations: distinct real roots
Homogeneous equations: repeated and complex roots
Repeated roots
Complex roots
Nonhomogeneous equations
Undetermined coefficients
Variation of parameters
Forced motion: beats and resonance
Higher order linear differential equations
Solving characteristic equations using Maple
For further study
Damped motion
Forced oscillations with damping
The Cauchy-Euler equation
Companion systems and companion matrices
Laplace transforms
Motivating problems
Laplace transforms: getting started
General properties of the Laplace transform
Piecewise continuous functions
The Heaviside functions
The Dirac delta function
The Heaviside and Dirac functions in Maple
Solving IVPs with the Laplace transform
More on the inverse Laplace transform
Laplace transforms and inverse transforms using Maple
For further study
Laplace transforms of infinite series
Laplace transforms of periodic forcing functions
Laplace transforms of systems
Nonlinear systems of differential equations
Motivating problems
Graphical behavior of solutions for 2 � 2 nonlinear systems
Plotting direction fields of nonlinear systems using Maple
Linear approximations of nonlinear systems
Euler's method for nonlinear systems
Implementing Euler's method for systems in Excel
For further study
The damped pendulum
Competitive species
Numerical methods for differential equations
Motivating problems
Beyond Euler's method
Heun's method
Modified Euler's method
Higher order methods
Taylor methods
Runge-Kutta methods
Methods for systems and higher order equations
Euler's methods for systems
Heun's method for systems
Runge-Kutta method for systems
Methods for higher order IVPs
For further study
Predator-Prey equations
Competitive species
The damped pendulum
Series solutions for differential equations
Motivating problems
A review of Taylor and power series
Power series solutions of linear equations
Legendre's equation
Three important examples
The Hermite equation
The Laguerre equation
The Bessel equation
The method of Frobenius
For further study
Taylor series for first-order differential equations
The Gamma function
Review of integration techniques
Complex numbers
Roots of polynomials
Linear transformations
Solutions to selected exercises
Index

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