System of Eq. – Distinct and Real eigenvalues
April 16, 2011
1
Introduction/Motivation
Suppose we’re given the follow
system
of differential equations and asked to solve it:
Note: here
x
is the
dependent
variable,
and
t
is the
indepen
dent
variable
x
0
1
(
t
) = 1
x
1
(
t
) + 1
x
2
(
t
)
(1)
x
0
2
(
t
) = 4
x
1
(
t
) + 1
x
2
(
t
)
(2)
If
x
2
(
t
)
did not appear in the first equation, we’d have a simple firstorder equation, and
could certainly solve it.
Therefore, it is only reasonable to try and eliminate
x
2
(
t
)
from
Eq. (
2
). Using Eq. (
1
), we find
x
2
(
t
) =
x
0
1
(
t
)

x
1
(
t
)
.
(3)
Plugging this expression into Eq. (
2
) we have
(
x
0
1
(
t
)

x
1
(
t
))
0
= 4
x
1
(
t
) + 1 (
x
0
1
(
t
)

x
1
(
t
))
.
Expanding the lefthandside:
x
00
1
(
t
)

x
0
1
(
t
) = 4
x
1
(
t
) +
x
0
1
(
t
)

x
1
(
t
)
,
or finally,
x
00
1

x
0
1

3
x
1
= 0
.
(4)
We can easily find the solution to above equation. The answer ends up being
x
1
(
t
) =
c
1
e
3
t
+
c
2
e

t
.
Using above expression and Eq. (
3
), we can easily solve
for
x
2
(
t
)
. The result turns out to
!
We
won’t
absorb the
2’s into the
c
1
and
c
2
here.
be
x
2
(
t
) = 2
c
1
e
3
t

2
c
2
e

t
.
(5)
To make a few observations, we will rewrite our results in the following vector form:
x
1
x
2
=
c
1
e
3
t
2
c
1
e
3
t
+
c
2
e

t

2
c
2
e

t
=
c
1
e
3
t
2
e
3
t
+
c
2
e

t

2
e

t
=
c
1
1
2
e
3
t
+
c
2
1

2
e

t
Key observations:
1
c HF, 2011
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System of Eq. – Distinct and Real eigenvalues
April 16, 2011
1. We started with a system of
first
order equations, and ended up with a
second
order
equation.
2. Our solution is in the following form:
x
1
x
2
=
ξ
1
ξ
2
e
rt
,
where
ξ
1
and
ξ
2
and constants.
In a more compact form, above equation may be
written as
x
(
t
) =
ξ
e
rt
,
(6)
where
x
(t) (bold) denotes a vectorvalued function.
This should motivate us to seek
two
solutions having the form of above equation. In other
words, we seek a general solution of the form
x
(
t
) =
c
1
ξ
(1)
e
r
1
(
t
)
+
c
2
ξ
(2)
e
r
2
(
t
)
.
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 Equations, Elementary algebra, general solution, real eigenvalues

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