Differential Equations and Matrix Algebra II (MA 222), Winter Quarter, 1999-2000
Exam 2 (100 points)
Name
Be neat and give complete answers. No notes, no text, no Maple.
2
5
12
5
3
5
1
20 pts 1. Let A =
with eigensystem
↔ −2,
−4
− 75
of graphs are related to the system of differential equations.
x
y
=
2
5
12
5
3
5
− 75
x
y
1
1
Box
↔ 1. The following set
4
y2
-4
-2
0
2
x
4
-2
-4
(a) Put the correct letter in the space provided.
Objects
The line x = 0
Choices
A
B
C
D
None of above
The line y = 0
E—line for λ = 1
E—line for λ = −2
(b) Indicate by writing C (correct) or I (incorrect) as to whether the paths P1, P2, P3, and
P4 are correctly drawn.
P1
P2
P3
P4
16 pts 2. Given the non—linear system of differential equations
x = −x3 + y
y = y + x2 − 2
(a) Verify that (1, 1) is a critical point.
(b) Linearize the above non—linear system at (1, 1). Put your answer in matrix/vector form
in the box below.
Answer:
x
y
=
(c) Given that the eigensystem of the matrix in (b) is
1√
2+ 6
√
↔ −1 + 6,
1√
2− 6
↔ −1 −
√
6
check the appropriate boxes to indicate the stability of the critical point (1, 1).
Asym Stable Stable Unstable Don’t Know
LS
NLS
16 pts 3. Given that the eigensystem for
−7 18
−3 8
is
3
1
↔ −1,
2
1
↔ 2,
(a) Find the general solution to
x
y
=
−7 18
−3 8
x
y
+
−1
−1
Put your answer in the box below.
General Solution:
x
y
=
gen
(b) What is the critical point for the above linear system?
(c) Check the appropriate box to indicate the stability of the critical point for the above
linear system.
Asym Stable Stable Unstable Don’t Know
LS
16 pts 4. Consider the second order differential equation
d2 θ
dθ
+ 3 + 2 sin(θ) = 0
2
dt
dt
which describes the motion of the pendulum.
(a) Circle the correct adjective: The above is a linear or non—linear differential equation.
(b) What does θ represent?
(c) Convert this differential equation to a Þrst order system (put your answer in the box
below).
Answer:
dx1
=
dt
dx2
=
dt
(d) Describe the actual motion of the pendulum corresponding to the following curve (from
A to B) in the phase plane. That is, where is the pendulum initially, which direction does it
go, etc.
x = f (x, y)
and A is
y = g(x, y)
the matrix obtained in the linearization process. Check the appropriate box which indicates
stability of the critical point (a, b). (LS = linearized system and NLS = non—linear system)
12 pts 5. Suppose that (a, b) is a critical point for some non—linear system
i) Eigenvalues of A are λ = 2 ± 3i.
Asym Stable Stable Unstable Don’t Know
LS
NLS
ii) Eigenvalues of A are λ = −1 and λ = −2.
Asym Stable Stable Unstable Don’t Know
LS
NLS
iii) Eigenvalues of A are λ = ±5i.
Asym Stable Stable Unstable Don’t Know
LS
NLS