Suppose that a certain population has a growth rate that varies with time and th

Question

Suppose that a certain population has a growth rate that varies with time and that this population satisfies the differential equation dy/dt = (0.5 + sin t)y/5. If y(0) = 1, find (or estimate) the time tau at which the population has doubled. Choose other initial conditions and determine whether the doubling time tau depends on the initial population. Suppose that the growth rate is replaced by its average value 1/10. Determine the doubling time tau in this case. Suppose that the term sin t in the differential equation is replaced by sin 2pi t; that is, the variation in the growth rate has a substantially higher frequency. What effect does this have on the doubling time tau? Plot the solutions obtained in parts (a), (b), and (c) on a single set of axes.

Explanation / Answer

5dy/y = (0.5 +sint ) dt 5 lny = 0.5t - cost + c y = e^((0.5t - cost)+c/5) put t =0 to get y(0)=1 we get c-1/5 =0 c=1 so y(t) = e^((0.5t - cost + 1)/5) for a) 5 lny = 0.5t - cost + c y2/y1 = 2 so 5 ln2 = (0.5T - cosT)+1 T= 6.9169 b)dy/dt = 1/10 y = t/10 +c y2 -y1 = T/10 T/10 = 2y1-y1 = y1 =1 T=10 c) 5dy/y = (0.5 +sin 2pit ) dt 5ln2 = (0.5t - cos 2pit / 2pi) + 1/2pi t=6.8459

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