642:527
SOLUTIONS: EXAM 2
FALL 2009
1. (a) Find the general solution of
z
′
=
A
z
, where
z
=
bracketleftbigg
x
y
bracketrightbigg
and
A
=
bracketleftbigg
−
4
4
1
−
4
bracketrightbigg
.
Be sure you
actually give this solution
(which should involve two free constants).
(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
+ 8
λ
+ 12 = (
λ
+ 2)(
λ
+ 6) the eigenvalues are
λ
1
=
−
2,
λ
2
=
−
6. The corresponding eigenvectors
z
(
i
)
are found by solving (
A
−
λ
i
)
z
(
i
)
=
0
:
λ
1
=
−
2 :
bracketleftbigg
−
2
4
1
−
2
bracketrightbigg bracketleftbigg
x
y
bracketrightbigg
=
0
=
⇒
z
(1)
=
bracketleftbigg
2
1
bracketrightbigg
λ
2
=
−
6 :
bracketleftbigg
2
4
1
2
bracketrightbigg bracketleftbigg
x
y
bracketrightbigg
=
0
=
⇒
z
(2)
=
bracketleftbigg
−
2
1
bracketrightbigg
The general solution is
z
(
t
) =
c
1
e
−
2
t
z
(1)
+
c
2
e
−
6
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 toward the origin.
The straight line trajectories are parallel to
the eigenvectors found in (a). The other tra
jectories are determined by the fact that as
as
t
→∞
they are parallel to
z
(1)
and as
t
→−∞
to
z
(2)
.
2. Consider the system
x
′
=
y
−
1
,
y
′
=
y
−
x
2
.
(a) Determine its singular (equilibrium) points find the linearized system near each.
For each
singular point, determine
the type of its linearized behavior (focus, node, etc.)
and whether
it is
unstable, stable, or asymptotically stable
for the linearized problem.
(b) Answer the same questions—type, stability—about the nature of these singular points in the
original, nonlinear system, if you have the information to do so, or explain why you cannot.
(c) Sketch the phase plane for this system, showing nullclines (curves on which the solution curves
have horizontal or vertical tangents), direction of motion as the solutions cross the nullclines, and
direction of motion in the regions separated by the nullclines.
No trajectories need be sketched.
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 Fall '08
 Staff
 Math, Fourier Series, Boundary value problem, Partial differential equation, ... ...

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