ð _
x t
ð Þ _
y t
ð ÞÞ ¼ x y
ð ÞUU
{ 1 2
2 1
UU
{ ,
ð15:99Þ
where U is a unitary matrix; i.e., U
{
¼ U
À1 .
Following routine calculation procedures (as in, e.g., Example 14.3), we readily
get a following diagonal matrix and the corresponding unitary matrix for the
diagonalization. That is, we obtain
U ¼
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
À
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
0
B
B
@
1
C
C
A and D ¼ U
À1 1 2
2 1
U ¼ U
À1 AU
¼
À1 0
0 3
:
ð15:100Þ
Or we have
A ¼ UDU
À1
:
ð15:101Þ
Hence, we get
F t
ð Þ ¼ exp tA ¼ exp tUDU
À1
À
Á ¼ U exp tD
ð
ÞU
À1 ,
ð15:102Þ
where with the last equality we used (15.30). From Theorem 15.3, we have
exp tD ¼
e
Àt
0
0 e
3t
:
ð15:103Þ
In turn, we get
F t
ð Þ ¼
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
À
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
0
B
B
@
1
C
C
A
e
Àt
0
0 e
3t
! 1
ffiffi ffi
2
p
À
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
0
B
B
@
1
C
C
A
¼
1
2
e
Àt
þ e
3t
Àe
Àt
þ e
3t
Àe
Àt
þ e
3t
e
Àt
þ e
3t
!
:
ð15:104Þ
With an inverse matrix, we have
F t
ð Þ
À1 ¼
1
2
e
À3t
þ e
t
e
À3t
À e
3t
e
À3t
À e
3t
e
À3t
þ e
t
:
ð15:105Þ
15.3 System of Differential Equations
601