1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 373

Cancelling the nonzero scalar factor erx, we find the unknowns v and r must

satisfy Av= rv. That is, for (14) y = v erx to be a solution of (13), r must

be an eigenvalue of A and v must be an associated eigenvector. Hence, we

immediately have the following theorem.

A THEOREM ON THE GENERAL SOLUTION OF

HOMOGENEOUS LINEAR SYSTEMS

If r1, r2, ... , rn a re the eigenvalues (not necessarily distinct) of an n x n

constant matrix A and if v 1 , v 2 ,... , v n are associated linearly independent

eigenvectors, then the general solution of t h e homogeneous linear system

y' = Ay is

(15)

where c 1 , c 2 , ... , Cn are arbitrary constants.

If the eigenvalues of A are distinct, then t here are n linearly independent
eigenvectors- one eigenvector corresponding to each eigenvalue. Moreover, if

each eigenvalue of m ultiplicity m > 1 has m associated linearly independent

eigenvectors, then there are a total of n linearly independent eigenvectors. In
either of these cases, we can use the computer software to find t he eigenvalues
and associated linearly independent eigenvectors and thereby write the general

solution of (13) y' = Ay in t he form of equation (15). It is only when there

is some eigenvalue of multiplicity m > 1 with fewer than m linearly indepen-

dent associated eigenvectors that we will not be able to write the solution of
(13) y' = Ay in the form of equation (15). In su ch a case, other techniques
must be used to find the general solution of (13). These techniques will not
be discussed in this text.

EXAMPLE 3 Finding the General Solution of a Homogeneous

Linear System

Find t h e general solution of the homogeneous linear system

(16) y' ~ G ~ ~) y

SOLUTION

We ran EIGEN by setting the size of the matrix equal to 3 and entering
the values for the elements of the given matrix. The output of the program
is displayed in Figure 8.6.