Motivated by the authors' combined ability and experience, this book is about the concepts of mathematical modelling with the use of differential equations, as a powerful technique of mathematical analysis. It is both enjoyable to read, and informative. The reader's mind is continually exercised by enlightenment, or recollection, or enquiry; either something new is to be learned, or something known is to be re-examined. It is with mathematical clarity that the authors explain the theory of ordinary differential equations and introduce their manifold applications. They show a skillful and imaginative succession of applications introducing such instances as accident risk or fish population; forgery detection in old masters or kidney life-support machines; and many others. They show the influence of these mathematical probes into problems occurring in biology, economics, geography, medicine, planning, psychology, or sociology. Introduction -- Growth and decay--the differential equation' dy [over] dx = ky' -- Variables separable differential equations -- Linear first order differential equations -- Linear second order differential equation -- Non-linear second order differential equations -- Systems of differential equations.