Calculus and Differential Equations with MATLAB Calculus and Differential Equations with MATLAB Pramote Dechaumphai α Chulalongkorn University Press Alpha Science International Ltd. Bangkok Oxford, U.K. Calculus and Differential Equations with MATLAB 454 pgs. | 169 figs. Pramote Dechaumphai Department of Mechanical Engineering Chulalongkorn University, Bangkok Copyright © 2016 ALPHA SCIENCE INTERNATIONAL LTD. 7200 The Quorum, Oxford Business Park North Garsington Road, Oxford OX4 2JZ, U.K. www.alphasci.com ISBN 978-1-78332-265-7 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the publisher. There are so many kinds of books for me to read. Some bring me joy; Some give me knowledge and entertainment. I therefore read every day. Books present a diversity of subjects, For readers to explore for their own pleasure. Since there is so much that can be discovered in books, I will continue reading for the rest of my life without getting bored. —HRH Princess Maha Chakri Sirindhorn, composed when she was 12 years old Translated by the Chalermprakiat Center of Translation and Interpretation, Faculty of Arts, Chulalongkorn University, as part of Chulalongkorn University Press’s celebration of the 60th Birthday of HRH Princess Maha Chakri Sirindhorn Preface MATLAB is a software widely used to solve mathematical problems that arise in the fields of science and engineering. These problems require profound understanding in the fundamentals of calculus and differential equations. MATLAB can solve many types of calculus problems and differential equations symbolically for exact closed-form expressions. If their exact solutions are not available, approximate solutions are obtained by using numerical methods. This book, Calculus and Differential Equations with MATLAB, explains how to use MATLAB to solve calculus and differential equation problems in a clear and easy-to-understand manner. Essential topics in the calculus and differential equation courses are selected and presented. These topics include: limit, differentiation, integration, series, special functions, Laplace and Fourier transforms, ordinary and partial differential equations. Numerous examples are used to present detailed derivation for their solutions. These solutions are carried out by hands as normally done in classes and verified by using the MATLAB commands. Students thus understand detailed mathematical process for obtaining solutions. They, at the same time, realize the software capability that can provide the same solutions effectively within a short time. The solutions are then plotted to provide clear understanding of their behaviors. The author would like to thank Miss Kaiumporn Phong-khachorn for her fine typing and his graduate students for proof-reading the manuscript. He would like to thank Chulalongkorn University Press and Alpha Science International for printing and distributing this book. He appreciates his wife Mrs. Yupa Dechaumphai for her support while writing this book. Pramote Dechaumphai Contents Preface vii 1 Symbolic Mathematics by MATLAB 1 1.1 Introduction 1 1.2 Symbolic Mathematics Software 2 1.3 History and Capability of MATLAB 5 1.4 MATLAB Fundamentals 6 1.5 Concluding Remarks 12 Exercises 13 2 Calculus 19 2.1 Introduction 19 2.2 Limits 20 2.3 Differentiation 26 2.4 Integration 36 2.5 Taylor Series 43 2.6 Other Series 48 2.7 Concluding Remarks 50 Exercises 51 3 Differential Equations 57 3.1 Introduction 57 3.2 Characteristics of Differential Equations 58 3.3 Solutions of Differential Equations 60 3.4 Concluding Remarks 70 Exercises 70 4 First-Order Differential Equations 79 4.1 Introduction 79 4.2 Separable Equations 80 4.3 Linear Equations 88 4.4 Exact Equations 94