Page 308 - A Course in Linear Algebra with Applications

292 Chapter Eight: Eigenvectors and Eigenvalues

Thus we are faced with the linear system of differential equa-
tions
y' =-2y + z
z' = y - 2z

Here
A=l~l _X)^Y=(l

Now the matrix A is diagonalizable; indeed

1 l l
D = S~ AS=[~ _° 3 J where S = ( _J

X
Setting U = S~ Y, we obtain from Y' = AY the equation
U' = DU. This yields two very simple differential equations

u[ = — u\
u'o = —3u
2
where u\ and «2 are the entries of U. Hence u\ = ce * and
3t
u-i = de~ , with arbitrary constants c and d. Finally

ce * + de 3t
Y = SU- l a _ t _ . _ 3 t
t
ce~ — de
The general solution of the original system of differential equa-
tions is therefore

y = ce~* + de~ 3t

z = ce~ f — de~ 3t

Thus the temperatures of both regions A and B tend to zero
as t —> oo.
In the next example complex eigenvalues arise, which
causes a change in the procedure.