TABLE OF CONTENTS The present chapter serves as an introduction. The first section contains several historical comments, while the second one is dedicated to a general presentation of the discipline. The third section reviews the most representative differential equations which can be solved by elementary methods. In the fourth section we gathered several mathematical models which illustrate the applicative power of the discipline. The fifth section is dedicated to some integral inequalities which will prove useful later, while the last sixth section contains several exercises and problems (whose proofs can be found at the end of the book).
The name of equatio differentialis has been used for the first time in 1676 by Gottfried Wilhelm von Leibniz in order to designate the determination of a function to satisfy together with one or more of its derivatives a given relation. This concept arose as a necessity to handle into a unitary and abstract frame a wide variety of problems in Mathematical Analysis and Mathematical Modelling formulated (and some of them even solved) by the middle of the XVII century. One of the first problems belonging to the domain of differential equations is the so-called problem of inverse tangents consisting in the determination of a plane curve by knowing the properties of its tangent at any point of it. The first who has tried to reduce thisproblem to quadratures [1] was Isaac Barrow [2] (1630 1677) who, using a geometric procedure invented by...
