Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

4.2

2 The differential equation used to model population growth, under the assumption that the proportional growth rate is a decreasing linear function of population, is the logistic differential equation dP d = rP(1-) where r and M are positive constants. By the change of variables, P(t) = Mx(t) we can obtain the simpler, related differential equation d = rx(1-x) for x > 0. (a) Express (1-2) in partial fraction form; hence find (1-2) dx, expressing your answer in the form In(). (Make sure you consider the two cases x < 1 and x > 1 separately.) (b) If x= xo> 0 when t = 0, show that the solution of the equation is X = 1 1+(-1) e-rt (c) Considering the two cases xo < 1 and xo > 1, determine what happens in the long run. (d) What happens if xo = 0 or xo = 1?
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Transcribed Image Text:2 The differential equation used to model population growth, under the assumption that the proportional growth rate is a decreasing linear function of population, is the logistic differential equation dP d = rP(1-) where r and M are positive constants. By the change of variables, P(t) = Mx(t) we can obtain the simpler, related differential equation d = rx(1-x) for x > 0. (a) Express (1-2) in partial fraction form; hence find (1-2) dx, expressing your answer in the form In(). (Make sure you consider the two cases x < 1 and x > 1 separately.) (b) If x= xo> 0 when t = 0, show that the solution of the equation is X = 1 1+(-1) e-rt (c) Considering the two cases xo < 1 and xo > 1, determine what happens in the long run. (d) What happens if xo = 0 or xo = 1?
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