By Sloughter D.
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Extra info for Difference equations to differential equations
Example text
4 We have seen examples of first-order linear equations in the population growth and radioactive decay examples above. Another interesting example arises in modeling the change in temperature of an object placed in an environment held at some constant temperature, such as a cup of tea cooling to room temperature or a glass of lemonade warming to room temperature. 9) n = 0, 1, 2, . , where k is a constant which depends upon the object. This difference equation is known as Newton’s law of cooling.
Example Suppose a certain population of owls is growing at the rate of 2% per year. 1) for n = 0, 1, 2, . .. That is, the number of owls in any given year is equal to the number of owls in the previous year plus 2% of the number of owls in the previous year. 1) is an example of a first-order difference equation; it relates the number of owls in a given year with the number of owls in the previous year. Hence we know the value of a specific xn once we know the value of xn−1 . To get the sequence started we have to know the value of x0 .
04yn , n = 0, 1, 2, . 04)n for n = 0, 1, 2, . .. 0 As we would expect, the population is growing more slowly under the inhibited population growth model than under the uninhibited model. Moreover, this difference will become more pronounced over time. 4 and y150 = 35, 892, illustrating how the inhibited growth model is constrained by the carrying capacity of 500 while the uninhibited growth model will have unbounded growth. 2 provide a graphical comparison of the two models for n = 0, 1, 2, . .