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Systems of Frstorder differential equations
161
We investigated this problem in the last section. What we wish to fnd is the
eigenvalues oF
A
and the associated eigenvectors. Return to the situation with only
two variables, and let the two roots (the two eigenvalues) be real and distinct,
which we shall again label as
r
and
s
. Let
v
r
be the eigenvector associated with the
root
r
and
v
s
be the eigenvector associated with the root
s
. Then so long as
r
±=
s
u
1
=
e
rt
v
r
and
u
2
=
e
st
v
s
are independent solutions, while
x
=
c
1
e
rt
v
r
+
c
2
e
st
v
s
(4.22)
is a general solution.
Example 4.8
±ind the general solution to the dynamic system
˙
x
=
x
+
y
˙
y
=−
2
x
+
4
y
We can write this in matrix Form
±
˙
x
˙
y
²
=
±
11
−
24
²±
x
y
²
The matrix
A
oF this system has already been investigated in terms oF example 4.7.
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 Fall '11
 Dr.Gwartney
 Economics

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