1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 377

an associated eigenvector, ).. and x satisfy Ax = >.x. Taking the complex

conjugate of this equation, we find (Ax) = (>.x). Since the conjugate of a

product equals the product of the conjugates, Ax = :\x. Because A has

real entries A = A , and we see that :\ and x satisfy Ax = :\x. That is,

:\ = a - if3 is an eigenvalue of A and x = u - iv is an associated eigenvec-

tor. Since f3 =f. 0, the eigenvalues ).. and :\ are distinct and, therefore, the

associated complex-valued solutions

and

w(x) = e<a-i,B)x(u - iv)= Y1(x) - iy2(x)

are linearly independent. Any two linear combinations of the solutions z(x)

and w(x), c1z(x) + c2w(x) and c3z(x) + c4w(x), where c 1 , c2, c3, and C4

are complex constants will also be linearly independent solutions provided

c 1 c 4 - c3c2 =f. 0. Choosing c 1 = c2 = 1/2, one linear combination is

And choosing c3 = -i/2 and c 4 = i/2, a second linear combination is

- i i -i i

2z(x) + 2w(x) = 2(Y1 + iy2) + 2(Y1 - iy2) = Y2(x).

Since c 1 c 4 - c3c2 = (l/2)(i/2) - (-i/2)(1/2) = i/2 =/. 0, the eigenvectors

y 1 (x) and Y2(x) are linearly independent solutions of y' = Ay.

Now, we use the theorem stated and proved above to produce a real general
solution of the homogeneous linear system

y I = Ay = (1 1 -2 3 ) y.

Earlier, we had found that one eigenvalue of the matrix A is)..= 2+i = a+f3i

and that the associated eigenvector is

(

Z 1 = -1-i i) = (-1) O + 2. (-1) l = U + ZV..

By the previous theorem, two real, linearly independent solutions of y' = Ay
are

( ) 2x (-1) 2x. (-1 ) (-e

2

x cos x + e

2

y x sin x)

1 x = e cosx 0 - e smx 1 = -e2xsinx