Purpose and scope

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.

How to use this book

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 (such as ∈δ-definitions of continuity and differentiability), 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.