Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.
This problem book contains exercises for courses in differential equations and calculus of variations at universities and technical institutes. It is designed for non-mathematics students and also for scientists and practicing engineers who feel a need to refresh their knowledge. The book contains more than 260 examples and about 1400 problems to be solved by the students — much of which have been composed by the authors themselves. Numerous references are given at the end of the book to furnish sources for detailed theoretical approaches, and expanded treatment of applications. Contents:First Order Differential EquationsN-th Order Differential EquationsLinear Second Order EquationsSystems of Differential EquationsPartial Equations of the First OrderNonlinear Equations and StabilityCalculus of VariationsAnswers to Problems Readership: Mathematicians and engineers. keywords:Examples;Differential Equations;Calculus of Variations “… the book can be successfully used both by students and practising engineers.” Mathematics Abstracts
Publisher: The Mathematical Association of America
This book presents a modern treatment of material traditionally covered in the sophomore-level course in ordinary differential equations. While this course is usually required for engineering students the material is attractive to students in any field of applied science, including those in the biological sciences. The standard analytic methods for solving first and second-order differential equations are covered in the first three chapters. Numerical and graphical methods are considered, side-by-side with the analytic methods, and are then used throughout the text. An early emphasis on the graphical treatment of autonomous first-order equations leads easily into a discussion of bifurcation of solutions with respect to parameters. The fourth chapter begins the study of linear systems of first-order equations and includes a section containing all of the material on matrix algebra needed in the remainder of the text. Building on the linear analysis, the fifth chapter brings the student to a level where two-dimensional nonlinear systems can be analyzed graphically via the phase plane. The study of bifurcations is extended to systems of equations, using several compelling examples, many of which are drawn from population biology. In this chapter the student is gently introduced to some of the more important results in the theory of dynamical systems. A student project, involving a problem recently appearing in the mathematical literature on dynamical systems, is included at the end of Chapter 5. A full treatment of the Laplace transform is given in Chapter 6, with several of the examples taken from the biological sciences. An appendix contains completely worked-out solutions to all of the odd-numbered exercises. The book is aimed at students with a good calculus background that want to learn more about how calculus is used to solve real problems in today's world. It can be used as a text for the introductory differential equations course, and is readable enough to be used even if the class is being "flipped." The book is also accessible as a self-study text for anyone who has completed two terms of calculus, including highly motivated high school students. Graduate students preparing to take courses in dynamical systems theory will also find this text useful.
An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences
Author: Morris Tenenbaum,Harry Pollard
Publisher: Courier Corporation
Skillfully organized introductory text examines origin of differential equations, then defines basic terms and outlines the general solution of a differential equation. Subsequent sections deal with integrating factors; dilution and accretion problems; linearization of first order systems; Laplace Transforms; Newton's Interpolation Formulas, more.
This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm-Liouville boundary value problems, including oscillation theory, are investigated. The second part introduces the concept of a dynamical system. The Poincare-Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems. The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits. The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.
This monograph explores various aspects of the inverse problem of the calculus of variations for systems of ordinary differential equations. The main problem centres on determining the existence and degree of generality of Lagrangians whose system of Euler-Lagrange equations coicides with a given system of ordinary differential equations. The authors rederive the basic necessary and sufficient conditions of Douglas for second order equations and extend them to equations of higher order using methods of the variational bicomplex of Tulcyjew, Vinogradov, and Tsujishita. The authors present an algorithm, based upon exterior differential systems techniques, for solving the inverse problem for second order equations. a number of new examples illustrate the effectiveness of this approach.
This introductory course in ordinary differential equations, intended for junior undergraduate students in applied mathematics, science and engineering, focuses on methods of solution and applications rather than theoretical analyses. Applications drawn mainly from dynamics, population biology and electric circuit theory are used to show how ordinary differential equations appear in the formulation of problems in science and engineering. The calculus required to comprehend this course is rather elementary, involving differentiation, integration and power series representation of only real functions of one variable. A basic knowledge of complex numbers and their arithmetic is also assumed, so that elementary complex functions which can be used for working out easily the general solutions of certain ordinary differential equations can be introduced. The pre-requisites just mentioned aside, the course is mainly self-contained. To promote the use of this course for self-study, suggested solutions are not only given to all set exercises, but they are also by and large complete with details.
The aim of the present book is to give a systematic treatment of the inverse problem of the calculus of variations, i.e. how to recognize whether a system of differential equations can be treated as a system for extremals of a variational functional (the Euler-Lagrange equations), using contemporary geometric methods. Selected applications in geometry, physics, optimal control, and general relativity are also considered. The book includes the following chapters: - Helmholtz conditions and the method of controlled Lagrangians (Bloch, Krupka, Zenkov) - The Sonin-Douglas's problem (Krupka) - Inverse variational problem and symmetry in action: The Ostrogradskyj relativistic third order dynamics (Matsyuk.) - Source forms and their variational completion (Voicu) - First-order variational sequences and the inverse problem of the calculus of variations (Urban, Volna) - The inverse problem of the calculus of variations on Grassmann fibrations (Urban).
This text explores the essentials of partial differential equations as applied to engineering and the physical sciences. Discusses ordinary differential equations, integral curves and surfaces of vector fields, the Cauchy-Kovalevsky theory, more. Problems and answers.
This book presents a method for solving linear ordinary differential equations based on the factorization of the differential operator. The approach for the case of constant coefficients is elementary, and only requires a basic knowledge of calculus and linear algebra. In particular, the book avoids the use of distribution theory, as well as the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The case of variable coefficients is addressed using Mammana’s result for the factorization of a real linear ordinary differential operator into a product of first-order (complex) factors, as well as a recent generalization of this result to the case of complex-valued coefficients.
Linear Ordinary Differential Equations, a text for advanced undergraduate or beginning graduate students, presents a thorough development of the main topics in linear differential equations. A rich collection of applications, examples, and exercises illustrates each topic. The authors reinforce students' understanding of calculus, linear algebra, and analysis while introducing the many applications of differential equations in science and engineering. Three recurrent themes run through the book. The methods of linear algebra are applied directly to the analysis of systems with constant or periodic coefficients and serve as a guide in the study of eigenvalues and eigenfunction expansions. The use of power series, beginning with the matrix exponential function leads to the special functions solving classical equations. Techniques from real analysis illuminate the development of series solutions, existence theorems for initial value problems, the asymptotic behavior solutions, and the convergence of eigenfunction expansions.
The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields,especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners. * Focuses on the most often used features of Mathematica for the beginning Mathematica user * New applications from a variety of fields, including engineering, biology, and physics * All applications were completed using recent versions of Mathematica
An accessible and hands-on approach to modeling and predicting real-world phenomena using differential equations A Workbook for Differential Equations presents an interactive introduction to fundamental solution methods for ordinary differential equations. The author emphasizes the importance of manually working through computations and models, rather than simply reading or memorizing formulas. Utilizing real-world applications from spring-mass systems and circuits to vibrating strings and an overview of the hydrogen atom, the book connects modern research with the presented topics, including first order equations, constant coefficient equations, Laplace transforms, partial differential equations, series solutions, systems, and numerical methods. The result is a unique guide to understanding the significance of differential equations in mathematics, science, and engineering. The workbook contains modules that involve readers in as many ways as possible, and each module begins with "Prerequisites" and "Learning Objectives" sections that outline both the skills needed to understand the presented material and what new skills will be obtained by the conclusion of the module. Detailed applications are intertwined in the discussion, motivating the investigation of new classes of differential equations and their accompanying techniques. Introductory modeling sections discuss applications and why certain known solution techniques may not be enough to successfully analyze certain situations. Almost every module concludes with a section that contains various projects, ranging from programming tasks to theoretical investigations. The book is specifically designed to promote the development of effective mathematical reading habits such as double-checking results and filling in omitted steps in a computation. Rather than provide lengthy explanations of what readers should do, good habits are demonstrated in short sections, and a wide range of exercises provide the opportunity to test reader comprehension of the concepts and techniques. Rich illustrations, highlighted notes, and boxed comments offer illuminating explanations of the computations. The material is not specific to any one particular software package, and as a result, necessary algorithms can be implemented in various programs, including Mathematica®, Maple, and Mathcad®. The book's related Web site features supplemental slides as well as videos that discuss additional topics such as homogeneous first order equations, the general solution of separable differential equations, and the derivation of the differential equations for a multi-loop circuit. In addition, twenty activities are included at the back of the book, allowing for further practice of discussed topics whether in the classroom or for self-study. With its numerous pedagogical features that consistently engage readers, A Workbook for Differential Equations is an excellent book for introductory courses in differential equations and applied mathematics at the undergraduate level. It is also a suitable reference for professionals in all areas of science, physics, and engineering.
Multivariable Calculus, Linear Algebra, and Differential Equations, Second Edition contains a comprehensive coverage of the study of advanced calculus, linear algebra, and differential equations for sophomore college students. The text includes a large number of examples, exercises, cases, and applications for students to learn calculus well. Also included is the history and development of calculus. The book is divided into five parts. The first part includes multivariable calculus material. The second part is an introduction to linear algebra. The third part of the book combines techniques from calculus and linear algebra and contains discussions of some of the most elegant results in calculus including Taylor's theorem in "n" variables, the multivariable mean value theorem, and the implicit function theorem. The fourth section contains detailed discussions of first-order and linear second-order equations. Also included are optional discussions of electric circuits and vibratory motion. The final section discusses Taylor's theorem, sequences, and series. The book is intended for sophomore college students of advanced calculus.
Author: Kendall Atkinson,Weimin Han,David E. Stewart
Publisher: John Wiley & Sons
A concise introduction to numerical methodsand the mathematical framework neededto understand their performance Numerical Solution of Ordinary Differential Equations presents a complete and easy-to-follow introduction to classical topics in the numerical solution of ordinary differential equations. The book's approach not only explains the presented mathematics, but also helps readers understand how these numerical methods are used to solve real-world problems. Unifying perspectives are provided throughout the text, bringing together and categorizing different types of problems in order to help readers comprehend the applications of ordinary differential equations. In addition, the authors' collective academic experience ensures a coherent and accessible discussion of key topics, including: Euler's method Taylor and Runge-Kutta methods General error analysis for multi-step methods Stiff differential equations Differential algebraic equations Two-point boundary value problems Volterra integral equations Each chapter features problem sets that enable readers to test and build their knowledge of the presented methods, and a related Web site features MATLAB® programs that facilitate the exploration of numerical methods in greater depth. Detailed references outline additional literature on both analytical and numerical aspects of ordinary differential equations for further exploration of individual topics. Numerical Solution of Ordinary Differential Equations is an excellent textbook for courses on the numerical solution of differential equations at the upper-undergraduate and beginning graduate levels. It also serves as a valuable reference for researchers in the fields of mathematics and engineering.
Superb, self-contained graduate-level text covers standard theorems concerning linear systems, existence and uniqueness of solutions, and dependence on parameters. Focuses on stability theory and its applications to oscillation phenomena, self-excited oscillations, more. Includes exercises.