Page 26 - Schaum's Outline of Differential Equations

CHAPTER 2

An Introduction

to Modeling

and Qualitative

Methods

MATHEMATICAL MODELS

Mathematical models can he thought uf as equations. In this chapter, and in other parts of ihc book (sec
Chapter 7. Chapter 14 and Chapter 31. for example), we will consider equations which model certain real-world
situations.
For example, when considering a simple direct current (DC) electrical circuit, the equation V = Rl modelfi
the voltage drop (measured in volts) aeross a resistor (measured in ohms), « here / is the current (measured in
amperes). This equation is called Ohm's Law. named in honor of G. S. Ohm (1787-1854), a German physicist,
Onee Constructed, some models can he used to predict main physical situations. Kor example, weather
forecasting, the growth of a tumor, or the outcome of a roulette wheel, ean all be connected with some form of
mathematical modeling.
In this chapter, we consider \ariables that are continuous and how differential equations can he used in
modeling. Chapter 34 introduces the idea of difference equations. These are equations in which we consider
discrete variables; that is, uiriables which can lake on only certain values, such as whole numbers. With few
modifications, everything presented about modeling with differential equations also holds true viith regard to
modeling with difference equations.

THE "MODELING CYCLE"

Suppose we have a real-life situation (we want tofind the amount of radio-active material in some element).
Research may be able to model this situation (in the form of a "very difficult" differential equation). Technology
may be used to help us solve the equation (computer programs give us an answer). The technological answers
are then interpreted or communicaied in light of the real-life situation (the amount of radio-active material).
Figure 2-1 illustrates this c\cle.

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