Page 315 - A Course in Linear Algebra with Applications

8.3: Applications to Systems of Linear Differential Equations 299

Example 8.3.5
Solve the second order system

= -2y 2 + y[ + 2y' 2
v'i = 2y x +2y[ 2/2

The system may be converted to a first order system by
introducing two new functions

2/3 = 2/i and y 4 = y' 2.

Thus y'{ = y' 3 and y 2 = y 4. The given system is therefore
equivalent to the first order system

' 2/i = 2/3
2/2=2/4
y' 3 = -2y 2 + y 3 + 2y 4
2
, 2/4 = 2/i + 2y 3 - y 4
The coefficient matrix here is

0 1
0 0 1
A =
-2 1 2
0 2 - 1 /

Its eigenvalues turn out to be 1, —1, 2, —2, with corresponding
eigenvectors

2
/ \ /
-1 1 -1
-2 2 -2
\ 1/ W V 2 /
Therefore, if S denotes the matrix with these vectors as its
1
columns, we have S~ AS = D, the diagonal matrix with di-
1
-
-
agonal entries 1, 1,2, 2 . Now write U = S~ Y. Then the