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Section 9.2
4.
We know that
e
3
it
= cos(3
t
) +
i
sin(3
t
). Therefore the trajectory is a
circle of radius one, drawn in the counterclockwise direction and the period
is given by 3
T
= 2
π
, so
T
=
2
π
3
.
6.
The eigenvalues are
λ
=
±
i
and the corresponding eigenvectors are
~v
=
±
3
±
i
5
¶
.
Therefore the general solution is given by:
~x
(
t
) =
c
1
e
it
±
3 +
i
5
¶
+
c
2
e

it
±
3

i
5
¶
.
If
c
1
=
c
2
= 1 then, using Euler’s equation we get:
~x
(
t
) = (cos(
t
) +
i
sin(
t
))
±
3 +
i
5
¶
+ (cos(
t
)

i
sin(
t
))
±
3

i
5
¶
=
±
6cos(
t
)

2sin(
t
)
10cos(
t
)
¶
.
16.
If
A
=
±
0 1
a b
¶
then the trace is
b
and the determinant is

a
. By
Fact 9.2.5, the zero is stable if both
a
and
b
are negative.
22.
The eigenvalues are
λ
1
= 3 and
λ
2
=
1
2
. The corresponding eigenvec
tors are
v
1
=
±
1

1
¶
,
v
2
=
±
0
1
¶
and since the system is discrete, it corresponds to phase plane VII.
26.
We proceed like above.
λ
1
= 1,
λ
2
=

2,
v
1
=
±
0
1
¶
,
v
2
=
±
1

1
¶
1
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View Full Document and since the system is continuous, it corresponds to phase plane V.
28.
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This homework help was uploaded on 02/23/2008 for the course MATH 2210 taught by Professor Pantano during the Fall '05 term at Cornell University (Engineering School).
 Fall '05
 PANTANO
 Linear Algebra, Algebra, Equations

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