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Unformatted text preview: Chapter 9 Exponential Growth and Decay: Differential Equations 9.1 Observations about the exponential function In a previous chapter we made an observation about a special property of the function y = f ( x ) = e x namely, that dy dx = e x = y so that this function satisfies the relationship dy dx = y. We call this a differential equation because it connects one (or more) derivatives of a function with the function itself. In this chapter we will study the implications of the above observation. Since most of the applications that we examine will be timedependent processes, we will here use t (for time) as the independent variable. Then we can make the following observations: 1. Let y be the function of time: y = f ( t ) = e t Then dy dt = e t = y With this slight change of notation, we see that the function y = e t satisfies the differential equation dy dt = y. v.2005.1  September 4, 2009 1 Math 102 Notes Chapter 9 2. Now consider y = e kt . Then, using the chain rule, and setting u = kt , and y = e u we find that dy dt = dy du du dt = e u k = ke kt = ky. So we see that the function y = e kt satisfies the differential equation dy dt = ky. 3. If instead we had the function y = e kt we could similarly show that the differential equation it satisfies is dy dt = ky. 4. Now suppose we had a constant in front, e.g. we were interested in the function y = 5 e kt . Then, by simple differentiation and rearrangement we have dy dt = 5 d dt e kt = 5( ke kt ) = k (5 e kt ) = ky. So we see that this function with the constant in front also satisfies the differential equation dy dt = ky. 5. The conclusion we reached in the previous step did not depend at all on the constant out front. Indeed, if we had started with a function of the form y = Ce kt where C is any constant, we would still have a function that satisfies the same differential equation. 6. While we will not prove this here, it turns out that these are the only functions that satisfy this equation. A few comments are worth making: First, unlike algebraic equations, (whose solutions are num bers), differential equations have solutions that are functions . We have seen above that depending on the constant k , we get either functions with a positive or with a negative exponent (assuming that time t > 0). This leads to the two distinct types of behaviour, exponential growth or exponen tial decay shown in Figures 9.1 and 9.2. In each of these figures we see a family of curves, each of which represents a function that satisfies one of the differential equations we have discussed. v.2005.1  September 4, 2009 2 Math 102 Notes Chapter 9 Figure 9.1: Functions of the form y = Ce kt for k > 0 represent exponentially growing solutions....
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 Spring '08
 LALALA
 Exponential Function, Equations

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