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Unformatted text preview: 10 The Art and Science of Modeling with First-Order Equations For some, modeling is the building of small plastic replicas of famous ships; for others, modeling means standing in front of cameras wearing silly clothing; for us, modeling is the process of developing sets of equations and formulas describing some process of interest. This process may be the falling of a frozen duck, the changes in a population over time, the consumption of fuel by a car traveling various distances, the accumulation of wealth by one individual or company, the cooling of a cup of coffee, the electronic transmission of sound and images from a television station to a home television, or any of a huge number of other processes affecting us. A major goal of modeling, of course, is to predict how things will turn out at some point of interest, be it a point of time in the future or a position along the road. Along with this, naturally, is often a desire to use the model to determine changes we can make to the process to force things to turn out as we desire. Of course, some things are more easily modeled mathematically than others. For example, it will certainly be easier to mathematically describe the number of rabbits in a field than to mathematically describe the various emotions of these rabbits. Part of the art of modeling is the determination of which quantities the model will deal with (e.g., number of rabbits instead of emotional states). Another part of modeling is the balancing between developing as complete a model as possible by taking into account all possible influences on the process as opposed to developing a simple and easy to use model by the use of simplifying assumptions and simple approximations. Attempting to accurately describe all possible influences usually leads to such a complicated set of equations and formulas that the model (i.e., the set of equations and formulas weve developed) is unusable. A model that is too simple, on the other hand, may lead to wildly inaccurate predictions, and, thus, would also not be a useful model. Here, we will examine various aspects of modeling using first-order differential equations. This will be done mainly by looking at a few illustrative examples, though, in a few sections, we will also discuss how to go about developing and using models with first-order differential equations more generally. 211 212 The Art and Science of Modeling with First-Order Equations 10.1 Preliminaries Suppose we have a situation in which some measurable quantity of interest (e.g.: velocity of a falling duck, number of rabbits in a field, gallons of fuel in a vehicle, amount of money in a bank, temperature of a cup of coffee) varies as some basic parameter (such as time or position) changes. For convenience, lets assume the parameter is time and denote that parameter by t , as is traditional. Recall that, if Q ( t ) = the amount of that measurable quantity at time t , then d Q dt = the rate at which...
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