1 - Chapter 9 Exponential Growth and Decay: Differential...

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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 time-dependent 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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1 - Chapter 9 Exponential Growth and Decay: Differential...

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