1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 375

(

.707107) (-.707107) (0)

y(x) = C1e^3 x 0.000000 + c2ex 0.000000 + c3ex 1

.707107 .707107 0

where c1, c2, and c3 are arbitrary constants.

You may find it more appealing to write the general solution in the form

where k1, k2, and k3 are arbitrary constants. This representation was ac-

complished by multiplying the eigenvectors v 1 and v 2 by the scalar constant

1/.707107 to obtain the eigenvectors

In part 4 of example 5 of section 8.2, we ran the computer program EI GEN

and found that the eigenvalues of the matrix

A=(~ -~)

are > 11 = 2 + i, .A 2 = 2 - i and the associated eigenvectors are

Z1 -(-1-i) - i

Therefore, the general solution of the homogeneous linear system y' = Ay

may be written as

where c 1 and c 2 are arbitrary constants. The function y(x) of equation (17)
is a complex-valued function of the real variable x. Since the original system
of differential equations, y' = Ay, is a real system (that is, since all of the
entries of the matrix A are all real numbers), it is desirable to write the general
solution of the system as a real-valued function of x, if possible. The following
theorem specifies conditions under which we can write two real-valued, linearly

independent solutions for y' = Ay when A has a pair of complex conjugate

eigenvalues.