Algebraic Solution of the Logistic Equation: As you have seen, it is possible to solve

Chapter 7, Problem 8

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Algebraic Solution of the Logistic Equation: As you have seen, it is possible to solve the logistic differential equation like that in algebraically. Suppose that Separating the variables and integrating gives In this problem you will learn how to integrate on the left side. Then you will solve the logistic equation in algebraically. a. The fraction in the left integral can be split into partial fractions like this: where A and B stand for constants. Using suitable algebra, find the values of A and B. b. Integrate the differential equation. Show that you can transform the integrated equation into where k is a constant related to the constant of integration. c. Solve the logistic equation from part d of algebraically. Transform the answer so that population is in terms of time. Use the initial condition P = 131.7 in 1940 (when t = 0) to evaluate k.d. Use the algebraic solution you found inpart c to predict the population in 1950,1960, 1970, 1980, and 1990. How well dothe approximate solutions found by Eulersmethod in part f of compare withthese exact solutions? How well do theexact solutions compare with the actualpopulation in these years? Write aparagraph that describes your observationsabout how well different mathematicalmodels agree with each other and abouthow well they fit data from the real world.For 923, Ona Nyland moves to anuninhabited island. Being lonely for company, sheimports some pet rabbits. The rabbits multiply andbecome a nuisance! So she imports some foxes tocontrol the rabbit population.