642:527
SOLUTIONS: EXAM 2
FALL 2007
1. (a) Find the general solution of
z
′
=
A
z
, where
z
=
bracketleftbigg
x
y
bracketrightbigg
and
A
=
bracketleftbigg
4
3
1
2
bracketrightbigg
.
(b) Give a careful drawing of the phase plane (
xy
plane) for this system, showing enough trajectories
to indicate qualitatively the motion in each region of the plane, as well as any “special” (straight
line) trajectories. Your trajectories should be marked with arrowheads giving the direction in which
the solution moves as
t
increases.
Solution:
(a) Since det(
A
−
λI
) =
λ
2
−
6
λ
+ 5 = (
λ
−
5)(
λ
−
1) the eigenvalues are
λ
1
= 1,
λ
2
= 5.
The corresponding eigenvectors
z
(
i
)
are found by solving (
A
−
λ
i
)
z
(
i
)
=
0
:
λ
1
= 1 :
bracketleftbigg
3
3
1
1
bracketrightbigg bracketleftbigg
x
y
bracketrightbigg
=
0
=
⇒
z
(1)
=
bracketleftbigg
−
1
1
bracketrightbigg
λ
2
= 5 :
bracketleftbigg
−
1
3
1
−
3
bracketrightbigg bracketleftbigg
x
y
bracketrightbigg
=
0
=
⇒
z
(2)
=
bracketleftbigg
3
1
bracketrightbigg
The general solution is
z
(
t
) =
c
1
e
t
z
(1)
+
c
2
e
5
t
z
(2)
. This is an
unstable node
.
(b) Here is a solution plot, from Maple.
I
don’t know how to get Maple to put arrow
heads on curves, but as is clear from the di
rection field or from the form of the solution,
all trajectories are oriented away from the ori
gin. The straight line trajectories are parallel
to the eigenvectors found in (a).
The other
trajectories are determined by the fact that
as as
t
→∞
they are parallel to
z
(2)
and as
t
→−∞
to
z
(1)
.
2. (a) Consider the system
x
′
= 1
−
y,
y
′
=
x
2
−
y.
Determine its singular (equilibrium) points and classify each, insofar as possible, using linearization:
for each equilibrium point, determine whether it is unstable, stable, or asymptotically stable, and
if it is a focus, node, or saddle, if you have the information to do so. If one cannot determine the
type from the linearization, say so.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 SPEER
 Fourier Series, Partial differential equation, ... ..., Joseph Fourier

Click to edit the document details