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6) n = 0, 1, 2, . , which we call the inhibited growth model, also known as the discrete logistic equation. This is an example of a nonlinear difference equation because if we multiply β 2 out the right-hand side of the equation we have a quadratic term, namely, M xn . 4; in fact, many nonlinear difference equations are not solvable in terms of the elementary functions of calculus. Hence we will not consider any methods for solving such equations, relying instead on computing specific solutions by iterating the equation using a calculator or, preferably, a computer.

A radioactive element is known to decay at the rate of 2% every 20 years. (a) If initially you had 165 grams of this element, how much would you have in 60 years? (b) What is the half-life of this element? (c) Suppose that the bones of a certain animal maintain a constant level of this element while the animal is living, but the element begins to decay as soon as the animal dies. If a bone of this animal is found and is determined to have only 10% of its original level of this element, how old is the bone?

Moreover, since xn β > 0, M we have xn+1 = xn + β xn (M − xn ) > xn M for all n. Hence the sequence {xn } is monotone increasing and bounded, and so must have a limit. In Problem 8 you will be asked to verify that this limit is in fact M , as appeared to be the case in the previous example. If β > 1, it may be the case that there are values of n for which xn > M , in which case β xn (M − xn ) < 0 M and, as a consequence, xn+1 < xn . 5. 003xn (500 − xn ) 500 for n = 0, 1, 2, . .. 5 Note how the values increase rapidly (as we should expect with such a large value for β) to above the carrying capacity of 500, but then oscillate about 500, with the oscillations diminishing in size.

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Analysis, Calculus. Difference-eq To Differential eq


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