# Differential Equations

**Topics:**Derivative, Partial differential equation, Differential equation

**Pages:**153 (29358 words)

**Published:**March 31, 2013

Lecture notes for MATH 2351/2352 (formerly MATH 150/151)

Jeffrey R. Chasnov

The Hong Kong University of Science and Technology

The Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon Hong Kong

c Copyright ○ 2009–2012 by Jeffrey Robert Chasnov This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

Preface

What follows are my lecture notes for Math 2351/2352: Introduction to ordinary differential equations/Differential equations and applications, taught at the Hong Kong University of Science and Technology. Math 2351, with two lecture hours per week, is primarily for non-mathematics majors and is required by several engineering and science departments; Math 2352, with three lecture hours per week, is primarily for mathematics majors and is required for applied mathematics students. Included in these notes are links to short tutorial videos posted on YouTube. There are also some links to longer videos of in-class lectures. It is hoped that future editions of these notes will more seamlessly join the video tutorials with the text. Much of the material of Chapters 2-6 and 8 has been adapted from the textbook “Elementary differential equations and boundary value problems” by c Boyce & DiPrima (John Wiley & Sons, Inc., Seventh Edition, ○2001). Many of the examples presented in these notes may be found in this book, and I apologize to the authors and publisher, with hope that I have not violated any copyright laws. The material of Chapter 7 is adapted from the textbook “Nonlinear dynamics and chaos” by Steven H. Strogatz (Perseus Publishing, c ○1994). All web surfers are welcome to download these notes, watch the YouTube videos, and to use the notes and videos freely for teaching and learning. I welcome any comments, suggestions or corrections sent by email to jeffrey.chasnov@ust.hk.

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Contents

0 A short mathematical review 0.1 The trigonometric functions . . . . . . . . . . . . . . 0.2 The exponential function and the natural logarithm 0.3 Definition of the derivative . . . . . . . . . . . . . . 0.4 Differentiating a combination of functions . . . . . . 0.4.1 The sum or difference rule . . . . . . . . . . . 0.4.2 The product rule . . . . . . . . . . . . . . . . 0.4.3 The quotient rule . . . . . . . . . . . . . . . . 0.4.4 The chain rule . . . . . . . . . . . . . . . . . 0.5 Differentiating elementary functions . . . . . . . . . 0.5.1 The power rule . . . . . . . . . . . . . . . . . 0.5.2 Trigonometric functions . . . . . . . . . . . . 0.5.3 Exponential and natural logarithm functions 0.6 Definition of the integral . . . . . . . . . . . . . . . . 0.7 The fundamental theorem of calculus . . . . . . . . . 0.8 Definite and indefinite integrals . . . . . . . . . . . . 0.9 Indefinite integrals of elementary functions . . . . . . 0.10 Substitution . . . . . . . . . . . . . . . . . . . . . . . 0.11 Integration by parts . . . . . . . . . . . . . . . . . . 0.12 Taylor series . . . . . . . . . . . . . . . . . . . . . . . 0.13 Complex numbers . . . . . . . . . . . . . . . . . . . 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 5 5 6 6 7 8

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1 Introduction to odes 11 1.1 The simplest type of differential equation . . . . . . . . . . . . . 11 2 First-order odes 2.1 The Euler method . . . . . . . . . . . . . . . . . . . . . 2.2 Separable equations . . . . . . . . . . . . . . . . . . . . 2.3 Linear equations . . . . . . . . . . . . . . ....

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