Stanley Duplessy
MMG525 Decision Techniques for Managers CA01
Assignment #2
This report will show possible decision options of an outdoor car wash in Bostons
Dorchester neighborhood. The 1st option is the car wash being open for a full day, 2nd option wil

6.2. COUPLED FIRST-ORDER EQUATIONS
77
The computation of in our present example proceeds from the differential
equations as follows:
1
1
1 2 2 1 = 1 (1 2 ) 2 ( 1 + 2 )
2
2
= (21 + 22 )
< 0.
And since < 0, the spiral rotation is clockwise, as shown in Fig.

72
CHAPTER 6. SYSTEMS OF EQUATIONS
For 1 = 3, and unknown eigenvector v1 , we have from (6.9)
211 + 21 = 0,
411 221 = 0.
Clearly, the second equation is just the first equation multiplied by 2, so only
one equation is linearly independent. This will alway

74
CHAPTER 6. SYSTEMS OF EQUATIONS
phase space diagram
2
1.5
1
x2
0.5
0
0.5
1
1.5
2
2
1.5
1
0.5
0
x1
0.5
1
1.5
2
Figure 6.2: Phase space diagram for example with two real eigenvalues of same
sign.
Therefore, the eigenvalues of A are 1 = 4, 2 = 1. Proceedi

6.2. COUPLED FIRST-ORDER EQUATIONS
73
phase space diagram
2
1.5
1
x2
0.5
0
0.5
1
1.5
2
2
1.5
1
0.5
0
x1
0.5
1
1.5
2
Figure 6.1: Phase space diagram for example with two real eigenvalues of opposite sign.
Therefore, the equivalent second-order linear homog

6.3. NORMAL MODES
81
Next, we determine the eigenvector associated with 22 :
21 + 22 = 0,
so
that 21 = 22 . The normal mode associated with the frequency 2 =
( + 2)/ thus follows a motion where 1 = 2 . Again referring to
Fig. 6.5, during this motion the

76
CHAPTER 6. SYSTEMS OF EQUATIONS
phase space diagram
2
1.5
1
x2
0.5
0
0.5
1
1.5
2
2
1.5
1
0.5
0
x1
0.5
1
1.5
2
Figure 6.3: Phase space diagram for example with complex conjugate eigenvalues.
and
)
1
Imcfw_v =
Im
(cos + sin )
(
)
1
sin
.
= 2
cos
2

84
CHAPTER 7. NONLINEAR DIFFERENTIAL EQUATIONS
The omitted terms in the Taylor series expansion are proportional to 2 , and
can be made negligible over a short time interval with respect to the kept term,
proportional to , by taking (0) sufficiently small

6.3. NORMAL MODES
79
If A has only a single linearly independent eigenvector v, then (6.13) can be
solved for w (otherwise, it cannot). Using A, and v of our present example,
(6.13) is the system of equations given by
(
) (
) (
)
1 1
1
1
=
.
1
1
2
1
The f

Chapter 7
Nonlinear differential
equations and bifurcation
theory
Reference: Strogatz, Sections 2.2, 2.4, 3.1, 3.2, 3.4, 6.3, 6.4, 8.2
We now turn our attention to nonlinear differential equations. In particular, we
study how small changes in the paramete

Stanley Duplessy
MMG525 Decision Techniques for Managers CA01
Assignment #4
For my survey, I wanted to find out from the population I surveyed, how many
individuals carry more than one device, and what their average age was. I had no target audience,
and

80
CHAPTER 6. SYSTEMS OF EQUATIONS
k
K
k
m
m
x1
x2
Figure 6.5: Coupled harmonic oscillators.
The equations for the coupled mass-spring system form a system of two secondorder linear homogeneous odes. In matrix form,
x = Ax, or explicitly,
(
)
(
) (
)
2
1