In the book Looking at History Through Mathematics, Rashevsky [Ra, pp. 103110] considers

Chapter 5, Problem 5.2.17

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In the book Looking at History Through Mathematics, Rashevsky [Ra, pp. 103110] considers a model for a problem involving the production of nonconformists in society. Suppose that a society has a population of x(t) 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. If the birth and death rates for all individuals are assumed to be the constants b and d, respectively, and if conformists and nonconformists mate at random, the problem can be expressed by the differential equations dx(t) dt = (b d)x(t) and dxn (t) dt = (b d)xn (t) + rb(x(t) xn (t)), where xn (t) denotes the number of nonconformists in the population at time t. a. Suppose the variable p(t) = xn (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 dp(t) dt = rb(1 p(t)). b. Assuming that p(0) = 0.01, b = 0.02, d = 0.015, and r = 0.1, approximate the solution p(t) from t = 0 to t = 50 when the step size is h = 1 year. c. Solve the differential equation for p(t) exactly, and compare your result in part (b) when t = 50 with the exact value at that time.

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