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MA 36600 LECTURE NOTES: MONDAY, APRIL 13
Then the general solution to the nonhomogeneous equation is in the form
x
(
t
)=
c
1
x
1
(
t
)+
c
2
x
2
(
t
)+
X
(
t
)
where
c
1
and
c
2
are constants. In terms of matrices, we see that the general solution is in the form
x
(
t
)=
c
1
x
(1)
(
t
)+
c
2
x
(2)
(
t
)+
x
(
p
)
(
t
)
,
where we have denoted the vectors
x
(1)
(
t
)=
°
x
1
(
t
)
x
°
1
(
t
)
±
,
x
(2)
(
t
)=
°
x
2
(
t
)
x
°
2
(
t
)
±
,
x
(
p
)
(
t
)=
°
X
(
t
)
X
°
(
t
)
±
.
We can express the general solution to the homogeneous equation in the form
x
(
c
)
(
t
)=
c
1
x
(1)
(
t
)+
c
2
x
(2)
(
t
)=
Ψ
(
t
)
c
where
Ψ
(
t
)=
°
x
1
(
t
)
x
2
(
t
)
x
°
1
(
t
)
x
°
2
(
t
)
±
,
c
=
°
c
1
c
2
±
.
Note that
W
(
t
)=det
Ψ
(
t
) is the Wronskian.
Fundamental Set of Solutions.
Consider the general system of Frst order linear equations
d
dt
x
=
P
(
t
)
x
+
g
(
t
)
.
Say that we can Fnd (
n
+ 1) matrix functions
x
(1)
(
t
)
,...,
x
(
n
)
(
t
)
,
x
(
p
)
(
t
) such that
i. The
x
(
k
)
are solutions to the homogeneous equation
d
dt
x
(
k
)
=
P
(
t
)
x
(
k
)
,k
=1
,
2
,...,n
.
ii. The
n
×
n
matrix
Ψ
(
t
)=
x
11
(
t
)
x
12
(
t
)
···
x
1
n
(
t
)
x
21
(
t
)
x
22
(
t
)
···
x
2
n
(
t
)
.
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue UniversityWest Lafayette.
 Spring '09
 EdrayGoins
 Matrices

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