The first order system , where and are m-dimensional vectors, is said to be linear if , where is an matrix and an m-dimensional vector; if in addition, , a constant matrix, the system is said to be linear with constant coefficients. We require the general solution of such a system

(9.1)

Let be the general solution of the corresponding homogeneous system

(9.2)

If is any particular solution of (9.1), then is the general solution of (9.1).

A set of m linearly independent solutions of (9.2), is said to form a fundamental system of (9.2), and the most general solution of (9.2) may be written as a linear combination of the members of the fundamental system. It is easily seen that , where is an m-dimensional vector, is a solution of (9.2) if , that is, if is an eigen value of A and is the corresponding eigen vector. We consider only the case where A possesses m distinct possibly complex, eigen values The corresponding eigen vectors are then linearly, independent, and it follows that the solutions form a fundamental system of (9.2). The most general solution of (9.1) is then