Cambridge: Cambridge University Press, 2011. — 132 p. — (AIMS Library of Mathematical Sciences).This book explains key concepts and methods in the ﬁeld of ordinary diﬀerential equations. It assumes only minimal mathematical prerequisites but, at the same time, introduces the reader to the way ordinary diﬀerential equations are used in current mathematical research and in scientiﬁc modelling. It is designed as a practical guide for students and aspiring researchers in any mathematical science - in which I include, besides mathematics itself, physics, engineering, computer science, probability theory, statistics and the quantitative side of chemistry, biology, economics and ﬁnance. The subject of diﬀerential equations is vast and this book only deals with initial value problems for ordinary diﬀerential equations. Such problems are fundamental in modern science since they arise when one tries to predict the future from knowledge about the present. Applications of diﬀerential equations in the physical and biological sciences occupy a prominent place both in the main text and in the exercises. Numerical methods for solving diﬀerential equations are not studied in any detail, but the use of mathematical software for solving diﬀerential equations and plotting functions is encouraged and sometimes required. The book should be useful for students at a range of levels and with a variety of scientiﬁc backgrounds, provided they have studied diﬀerential and integral calculus (including partial derivatives), elements of real analysis, complex numbers and linear algebra. It could serve as a textbook for a ﬁrst course on ordinary diﬀerential equations for undergraduates on a mathematics, science, engineering or economics degree who have studied the prerequisites listed above. The exercises form an integral part of the text, and the reader is strongly encouraged to attempt them all. They range in diﬃculty from routine to challenging, and some introduce or develop key ideas which are only mentioned in the main text. Solutions are not included, and it is quite likely that the reader will initially get stuck on some of the problems.Preface. First order diﬀerential equations. Systems and higher order equations. Second order equations and oscillations. Geometric methods. Projects. References. Index.
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