Page 309 - A Course in Linear Algebra with Applications

8.3: Applications to Systems of Linear Differential Equations 29o

Example 8.3.2
Solve the linear system of differential equations

( y[ = Vi ~ Vi
\y'2 = yi+y2

The coefficient matrix here is

A =

which has complex eigenvalues 1 + % and 1 — i; we are us-
ing the familiar notation % = \/—l here. The corresponding
eigenvectors are

0 and (~i*

respectively. Let S be the 2x2 matrix which has these vectors
1
as its columns; then S~ AS = D, the diagonal matrix with
X
diagonal entries 1 + i and 1 — i. If we write U = S~ Y, the
system of equations becomes U' = DU, that is,

tti = (1 +i)u\

u' 2 = (1 - i)u 2

where u\ and U2 are the entries of U.
1+ x
The first equation has the solution u\ = e( ^ , while the
second has the obvious solution u 2 = 0. Using these values
for u\ and U2, we obtain a complex solution of the system of
differential equations

_ _ ( i e W \
Y s u
y _ su - I ( ) I
e 1+i x
Of course we are looking for real solutions, but these are in
fact at hand. For the real and imaginary parts of Y will also