Math 33b, Quiz 6ac, March 4, 2008
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1. Find the solution, with initial condition
~x
(0) =
±
2
3
²
, to the system of diﬀerential
equations
d~x
dt
=
±

1

1
4

5
²
~x.
Solution.
The characteristic equation of the matrix is det(
A

λI
) = det
±

1

λ

1
4

5

λ
²
,
which is
λ
2
+ 6
λ
+ 9. The roots are
λ
=

3
,

3, so there is only one eigenvalue
λ
=

3.
We attempt to ﬁnd eigenvectors for
λ
=

3. Solving (
A

(

3)
I
)
~v
=
~
0, we have
±
2

1
4

2
²±
v
1
v
2
²
=
±
0
0
²
. Writing them out as equations, we have
2
v
1

v
2
= 0
4
v
1

2
v
2
= 0
As usual, the two equations are multiples of each other, so we need only ﬁnd a nonzero
solution to 2
v
1

v
2
= 0. If we choose
v
1
= 1, then
v
1
= 2 and so
~v
=
±
1
2
²
.
To write down the solution to the system of diﬀerential equations, we must ﬁnd a
pseudoeigenvector
~w
such that (
A

(

3)
I
)
~w
=
~v
.
Writing everything out, we want
±
2

1
4

2
²±
w
1
w
2
²
=
±
1
2
²
. Writing these out, we have
2
w
1

w
2
= 1
4
w
1

2
w
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 Spring '07
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 Equations, initial condition

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