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Various visual features are used to highlight focus areas. Complete illustrative diagrams are used to facilitate mathematical modeling of application problems. Readers are motivated by a focus on the relevance of differential equations through their applications in various engineering disciplines.
Studies of various types of differential equations are determined by engi- neering applications. Theory and techniques for solving differential equations are then applied to solve practical engineering problems.
Detailed step-by-step analysis is pre- sented to model the engineering problems using differential equations from physical principles and to solve the differential equations using the easiest possible method. Such a detailed, step-by-step approach, especially when applied to practical engineering prob- lems, helps the readers to develop problem-solving skills.
This book is suitable for use not only as a textbook on ordinary differential equa- tions for undergraduate students in an engineering program but also as a guide to self- study. It can also be used as a reference after students have completed learning the subject. He is the author of Dynamic Stability of Structures and has published numerous journal articles on dynamic stability, structural dynamics and random vibration, nonlinear dynamics and stochastic mechanics, reliability and safety analysis of engineering systems, and seismic analysis and design of engineering structures.
He has been teaching differential equa- tions to engineering students for almost twenty years. He received the Teaching Excel- lence Award in in recognition of his exemplary record of outstanding teaching, concern for students, and commitment to the development and enrichment of engineer- ing education at Waterloo.
He is the recipient of the Distinguished Teacher Award in , which is the highest formal recognition given by the University of Waterloo for a superior record of continued excellence in teaching. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
First published Printed in the United States of America A catalog record for this publication is available from the British Library. Includes bibliographical references and index. ISBN 1. Differential equations. Engineering mathematics. D45X54 Contents Preface. Motivating Examples 1 1 1. The Method of Separation of Variables 16 16 2. Vibration of a Single Degree-of-Freedom System 5.
Mathematical Modeling of Mechanical Vibrations 8. In general, modeling of the variation of a physical quantity, such as temperature, pressure, displacement, velocity, stress, strain, current, voltage, or concentration of a pollutant, with the change of time or location, or both would result in differential equations.
Similarly, studying the variation of some physical quantities on other physical quantities would also lead to differential equations. In fact, many engineering subjects, such as mechanical vibration or structural dynamics, heat transfer, or theory of electric circuits, are founded on the theory of differential equations.
It is practically important for engineers to be able to model physical problems using mathematical equations, and then solve these equations so that the behavior of the systems concerned can be studied. I have been teaching differential equations to engineering students for the past two decades.
Most, if not all, of the textbooks are written by mathematicians with little engineering background. For engineering students, it is more important to know the applications and techniques for solving application problems than to delve into the nuances of mathematical concepts and theorems.
Knowing the appropriate applications can motivate them to study the mathematical concepts and techniques. However, it is much more challenging to model an application problem using physical principles and then solve the resulting differential equations than it is to merely carry out mathematical exercises. Mathematicians are more interested if: 1 there are solutions to a differential equation or a system of differential equations; 2 the solutions are unique under a certain set of con- ditions; and 3 the differential equations can be solved.
Hence, a detailed step-by-step approach, especially applied to practical engineering problems, helps students to develop problem solving skills. Readers often miss the points of importance. Studies of various types of differential equations are motivated by engineering applications; the- ory and techniques for solving differential equations are then applied to solve practical engineering problems. This book could be used as a reference after students have completed learning the subject.
As a reference, it has to be reasonably comprehensive and complete. Detailed step-by-step analysis is presented to model the engineering problems using differential equations and to solve the differential equations. Various visual features, such as side-notes preceded by the symbol , different fonts and shades, are used to highlight focus areas.
This book is not only suitable as a textbook for classroom use but also is easy for self-study. As a textbook, it has to be easy to understand. For self-study, the presentation is detailed with all necessary steps and useful formulas given as side-notes. Scope This book is primarily for engineering students and practitioners as the main audience. It is suitable as a textbook on ordinary differential equations for under- graduate students in an engineering program.
Some basic general concepts of differential equations are then introduced. Chapter 4 studies linear ordinary differential equations.
Complementary solu- tions are obtained through the characteristic equations and characteristic numbers. Applications involving linear ordinary differential equations are presented in Chapter 5. Solutions of linear ordinary differential equations using the Laplace transform are studied in Chapter 6, emphasizing functions involving Heaviside step function and Dirac delta function.
Chapter 7 studies solutions of systems of linear ordinary differential equations. The method of operator, the method of Laplace transform, and the matrix method are introduced. Applications involving systems of linear ordinary differential equa- tions are considered in Chapter 8. In Chapter 9, solutions of ordinary differential equations in series about an ordinary point and a regular singular point are presented. Some classical methods, including forward and backward Euler method, im- proved Euler method, and Runge-Kutta methods, are presented in Chapter 10 for numerical solutions of ordinary differential equations.
In Chapter 11, the method of separation of variables is applied to solve partial differential equations. When the method is applicable, it converts a partial differ- ential equation into a set of ordinary differential equations. Flexural vibration of beams and heat conduction are studied as examples of application.
Solutions of ordinary differential equations using Maple are presented in Chapter However, it cannot replace learning and thinking, especially mathematical modeling. This will also help the development of insight into the problems and appreciation of the solution process. For this reason, solutions of ordinary differential equations using Maple is presented in the last chapter of the book instead of a scattering throughout the book.
There are more than enough materials for a one-term semester undergraduate course. Instructors can select the materials according to the curriculum. Drafts of this book were used as the textbook in a one-term undergraduate course at the University of Waterloo. Acknowledgments First and foremost, my sincere appreciation goes to my students. It is the students who give me a stage where I can cultivate my talent and passion for teaching.
It is for the students that this book is written, as my small contribution to their success in academic and professional careers. My undergraduate students who have used the draft of this book as a textbook have made many encouraging comments and constructive suggestions. My sincere appreciation goes to Mr.
Peter Gordon, Senior Editor, Engineering, Cambridge University Press, for his encouragement, trust, and hard work to publish this book.
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