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chap2 - 27 27 2 Difference equations and modularity 2.1...

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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 discrete-time mathematics (differ- ence equations); and to start investigating the simplest second-order 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 discrete-time 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 non-modular 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 discrete-time 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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