In a book titled Looking atHistory Through Mathematics, Rashevsky [Ra], pp. 103-110

Chapter 5, Problem 17

(choose chapter or problem)

In a book titled Looking atHistory Through Mathematics, Rashevsky [Ra], pp. 103-110, considers a model for a problem involving the production ofnonconformists in society. Suppose that a society has a population of x(0 individuals at time t, in years, and that all nonconformists who mate with other nonconformists have offspring who are also nonconformists, while a fixed proportion r of all other offspring are also nonconformist. Ifthe birth rates and death rates for all individuals are assumed to be the constants h and d, respectively, and if conformists and nonconformists mate at random, the problem can be expressed by the differential equations = {h - d)x(t) and - {h - d)xll{t) + rh(x{t) - xn(r)), dt dt where xn (?) denotes the number of nonconformists in the population at time t. a. Suppose the variable pit) = x(t)/x(t) is introduced to represent the proportion of nonconformists in the society at time t. Show that these equations can be combined and simplified to the single differential equation dt b. Assuming piO) = 0.01, b = 0.02, d = 0.015, and r = 0.1, approximate the solution pit) from t = 0 to t = 50 when the step size is /; = 1 year. c. Solve the differential equation for pit) exactly and compare your result in part (b) when t = 50 with the exact value at that time