TABLE OF CONTENTS This chapter is dedicated to the introduction and study of the concept of prime integral for a system of first-order differential equations. In the first two sections we present the main notions and results referring to this problem in the case of both autonomous and non-autonomous systems. The third section is concerned with the study of first-order linear and quasi-linear partial differential equations, while the fourth section contains a fundamental existence and uniqueness theorem with regard to the Cauchy problem for a class of first-order quasi-linear partial differential equations. In the fifth section we collect some specific properties of the so-called conservation law. The chapter ends with an Exercises and Problems section.
Let ? be a nonempty and open subset in 
, let 
be a continuous function and let us consider the autonomous system
| (6.1.1) |  |
In many specific situations, considerations of extra-mathematical nature based on the physical signification of the functions involved in (6.1.1), prove the existence of some functions of class C 1, 
which, although nonconstant on ?, are constant along the trajectories [1] of the system (6.1.1). Any family of functionally independent such functions could be of real help in obtaining information on the solutions of (6.1.1) which, in most cases, cannot be explicitly solved. Moreover, the larger is such a family, the bigger are the chances to solve (6.1.1) explicitly or, at least, to obtain crucial information on its solutions. This is because, from a...
