This chapter contains the most important results referring to the Cauchy problem governed by a system of n first-order linear differential equations with n unknown functions. In the first section we show that the set of all saturated solutions of such a homogeneous system is an n-dimensional vector space over . The second section is dedicated to the study of non-homogeneous systems. Here we included the celebrated variation of constants formula. In the third and fourth sections we present two methods of finding an algebraic basis in the space of all saturated solutions of a homogeneous system with constant coefficients. The aim of the fifth section is to rephrase the previously proved results in order to handle as particular case the n ^th-order linear differential equation, while the sixth section is dedicated to a simple method of solving explicitly such equations with constant coefficients. The chapter ends with a section of exercises and problems.

4.1 Homogeneous Systems. The Space of Solutions

Let and be continuous for i, j=1, 2, , n, and let us consider the system of first-order linear differential equations

(4.1.1)

which, with the notations

and

for , can be rewritten as a first-order vector differential equation

(4.1.2)

For the sake of simplicity, in all what follows, we will write the system (4.1.1) only in the form of a vector differential equation (4.1.2), and we will call it by extension first-order system of linear differential equations.