2
Difference equations and modularity
2.1
Modularity: Making the input like the output
17
2.2
Endowment gift
21
2.3
Rabbits
25
The goals of this chapter are:
•
to illustrate modularity and to describe systems in a modular way;
•
to translate problems from their representation as a verbal descrip
tion into their representation as discretetime mathematics (differ
ence equations); and
•
to start investigating the simplest secondorder system, the second
simplest module for analyzing and designing systems.
The themes of this chapter are modularity and the
representation
of ver
bal descriptions as discretetime mathematics. We illustrate these themes
with two examples, money in a hypothetical MIT endowment fund and
rabbits reproducing in a pen, setting up difference equations to represent
them. The rabbit example, which introduces a new module for building
and analyzing systems, is a frequent visitor to these chapters. In this chap
ter we begin to study how that module behaves. Before introducing the
examples, we illustrate what modularity is and why it is useful.
2.1
Modularity: Making the input like the output
A common but alas nonmodular way to formulate difference and differ
ential equations uses boundary conditions. An example from population
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18
2.1 Modularity: Making the input like the output
growth illustrates this formulation and how to improve it by making it
modular. The example is the population of the United States. The US pop
ulation grows at an annual rate of roughly 1%, according to the
World Fact
Book
[2], and the US population is roughly 300 million in 2007. What will
be the US population be in 2077 if the growth rate remains constant at 1%?
Pause to try 1.
What is the population equation and boundary con
dition representing this information?
The difference equation for the population in year
n
is
p
[
n
] = (
1
+
r
)
p
[
n
−
1
]
(population equation)
,
where
r
=
0.01
is the annual growth rate. The boundary condition is
p
[
2007
] =
3
×
10
8
(boundary condition)
.
To find the population in 2077, solve this difference equation with bound
ary condition to find
p
[
2077
]
.
Exercise 1.
What is p[2077]? How could you have quickly ap
proximated the answer?
You might wonder why, since no terms are subtracted, the population
equation is called a difference equation.
The reason is by analogy with
differential equations, which tell you how to find
f
(
t
)
from
f
(
t
−
∆t
)
, with
∆t
going to 0. Since the discretetime population equation tells us how to
find
f
[
n
]
from
f
[
n
−
1
]
, it is called a difference equation and its solution is
the subject of the calculus of finite differences. When the goal – here, the
population – appears on the input side, the difference equation is also a
recurrence relation
. What recurrence has to do with it is the topic of an
upcoming chapter; for now take it as pervasive jargon.
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 Spring '11
 DennisM.Freeman
 Recurrence relation, Fibonacci number, Generating function

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