Math 2940
Worksheet: 7.4, 7.5, 8.1, 8.3, 9.1, 9.3
December 3, 2009
1. Let
A
be an
n
×
n
matrix and
k
a scalar. Consider the following two systems:
d~x
dt
=
A~x
d~
c
dt
=
kA~
c
Show that if
~x
(
t
) is a solution to the ﬁrst system, then
~
c
(
t
) =
~x
(
kt
) is a solution of the second
system.
2. Consider the interaction of two species of animals in a habitat. We are told that the change
of the populations
x
(
t
) and
y
(
t
) can be modeled by the equations
dx
dt
= 1
.
4
x

1
.
2
y
dy
dt
= 0
.
8
x

1
.
4
y
where time
t
is measured in years.
(a) Sketch a phase portrait for this system. From the nature of the problem, we are interested
only in the ﬁrst quadrant.
(b) What will happen in the long term? Does the outcome depend on the initial populations?
If so, how?
3. Consider the dynamical system
d~x
dt
=
±

1

2
2

1
²
~x
(a) Find all real solutions of this system.
(b) Solve the sytem with initial condition
~x
(0) =
±
1

1
²
.
4. Find all the eigenvalues and ”eigenvectors” of the linear transformations and determine if the
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 Spring '06
 PANTANO
 Multivariable Calculus, Scalar, Singular value decomposition, Diagonal matrix, Orthogonal matrix, dt dy dt, dt dc dt

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