Data aequatione quotcunque fluentes
quantitae involvente fluxiones invenire
et vice versa. [“Given an equation
which involves the derivatives of one
or more functions, find the functions.”]
—Sir Isaac Newton to G.W. Leibniz,
October 24, 1676
Calculus was perfected by Newton to formulate and solve differential equations arising in mechanics, and the subject of ordinary differential equations has always played a central role in the development of the infinitesimal calculus. Isaac Newton made differential equations an almost universal tool to model the physical world, and the success was so impressive that it opened the doors for a great development of our understanding of the laws of Nature. Differential equations have been applied not only to the physical world (say, physics and astronomy), but also to much more "human" sciences such as economics, physiology, population dynamics, epidemiology and management sciences. They have even had an impact on philosophical questions like causality and determinism. The key role played by differential equations is synthesized as follows by the famous contemporary Russian mathematician Vladimir I. Arnold (1937–2010):
Differential equations are one of the basic tools of mathematics. They were first considered systematically by Sir Isaac Newton (1642–1727), although problems leading to differential equations had in fact arisen earlier. Before Newton, however, only such geniuses as Christiaan Huygens (1629–1695), president of the French Academy of Sciences, and Isaac Barrow (1630–1677), a mathematician and theologian, who was Newton’s teacher, could solve them. Today, thanks to Newton, many differential equations are solvable by college students and even school children.
Differential equations are of wide mathematical interest. They are used extensively as mathematical models in applications from the physical sciences, engineering, economics, and the biological sciences. This is reflected in the many elementary textbooks on the subject. At the same time, differential equations provide significant concrete examples for abstract mathematical theories, giving indications of profitable directions for research. There are a smaller number of more advanced texts directed at upper level undergraduates or at graduate students. Such texts usually concentrate on theory and proofs and commonly pay little attention to applications.
One or two generations ago, courses in differential equations treated them as an application of integral calculus. The object was to develop techniques for finding explicit solutions. The currently prevailing approach is to find ways of deducing properties of solutions even when explicit formulas are not available. This is one of the main purposes of Classical Methods in Ordinary Differential Equations. Existence and uniqueness theorems for solutions of initial value problems are, of course, proved, so that solutions of initial value problems can be discussed with the knowledge that they make mathematical sense even though there is no explicit formula. Existence and uniqueness proofs that use successive approximations give the possibility of constructing useful approximations to solutions. Linear algebra allows us to describe the structure of the family of all solutions of linear differential equations. The authors also analyze continuity and differentiability of solutions with respect to initial conditions and parameters, which is important for applications (stability properties) since it is desired that small changes in initial conditions or parameters arising, say, through measurement errors, will result in only small changes in solutions.
The book of Hastings and McLeod is a superb introduction to the modern theory of differential equations. All the fundamental concepts, such as the shooting method, heteroclinic or homoclinic orbits, vector fields, layers and spikes, dynamical systems, geometric perturbation theory, are presented here in a clear and rigorous fashion. These often remain in the shadows in the traditional coordinate based approach, but here they are center-stage. The authors do much more than just pay lip service to these concepts, as many other books do: they do their darndest to make sure the reader understands these concepts.
This book brings a new and innovative look to several areas of the theory of ordinary differential equations. It is certainly a very refreshing addition to the existing literature. Students of mathematics who have avoided (for various reasons) differential equations need this volume as an antidote. Potential differential geometers should not wait another day before starting to learn how their interests relate to other areas. This book is an accessible introduction to the modern theory of a very active research field that lies at the interplay between pure and applied mathematics. As a conclusion, this is a most informative, stimulating, and refreshing book!
Vicentiu D. Radulescu (http://inf.ucv.ro/~radulescu) is a Professorial Fellow at the Mathematics Institute of the Romanian Academy. He received both his Ph.D. and Habilitation at the Université Pierre et Marie Curie (Paris 6) under the coordination of Haim Brezis. His current interests are broadly in nonlinear PDEs and their applications. He wrote more than 150 papers and he published books with Oxford University Press, Cambridge University Press, Springer New York, Springer Heidelberg, Kluwer, and Hindawi.