Ang K. Differential Equations. Models and Methods 2005

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Ang K. Differential Equations. Models and Methods 2005

Textbook in PDF format Differential equations appear in many areas of mathematics and science, as well as in other fields of studies such as economics, engineering and even social science. They are an important tool for constructing mathematical models to help us understand our physical world. The use of differential equations in academic research and industry is not only widespread but also successful. This is evident in the fact that many solutions to complex real life problems are based on differential equations. Apart from the practical relevance, the study of differential equations is in itself a beautiful application of concepts and ideas of calculus. In fact, differential equations are sometimes seen as a logical extension of calculus, and it has even been suggested that calculus was developed so that one could apply it to differential equations. With the advent of high-speed computers and personal computers, concepts and skills involved in using differential equations in mathematical modelling can be made more accessible. This partly accounts for the popularity of differential equations as a modelling tool. While there are literally hundreds of books written on differential equations, this book is different from them in several ways. Many existing books on differential equations assume a fairly sophisticated level of competence in calculus at the university level. They may not be very suitable for students who have a basic knowledge of calculus, but wish to learn more about differential equations and their applications. This book is also very focussed in that it is specifically about first order differential equations, and their methods of solutions and applications. By restricting the content, we are able to go further and deeper into the important conceptual ideas as well as interesting applications of first order differential equations. Numerical and graphical techniques are introduced in this book. Many books on differential equations tend to make reference to computer tools such as Computer Algebra Systems, or CAS. Readers of these books will need to have access to CAS in order to get the most out of the them. In this book, we have avoided the need to use CAS, although a simple example is mentioned. In fact, most of the graphical or numerical techniques discussed may be implemented using readily accessible tools such as a spreadsheet or graphing utility. This makes it more convenient to appreciate the techniques because there is no need to learn (or buy!) expensive software like CAS. Finally, this book serves as an introductory text aimed primarily at students taking the new GCE ‘A’ Level H3 Mathematics. These are pre-university students who may have only a rudimentary working knowledge of calculus and are encountering differential equations for the first time. To the best of my knowledge, no such text is currently available for this group of students. It is not the intention of this book to cover all aspects of differential equations, or even every concept in first order differential equations. The objective is to provide sufficient material for the target audience and at the same time stretch a little beyond that which is required for H3 Mathematics. The instructor using this book can choose to omit topics such as Picard’s Iteration Method or parts of Chapter 6. Every section ends with an Exercise Set. These serve to consolidate the ideas discussed in the section and readers are strongly encouraged to attempt them before moving on to the next section. In most of the exercises on applications or mathematical modelling, attempts have been made to provide as realistic a context as possible. In some cases, the data provided are from real life experiments or sources. Despite efforts to ensure that this book is free of errors and mistakes, I am sure it remains possible that some errors may have escaped detection. I would be most grateful if these are brought to my attention so that they may be attended to in due course