The propagation of a single action in a large population (for example, drivers turning

Chapter 2, Problem 35

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The propagation of a single action in a large population (for example, drivers turning on headlights at sunset) often depends partly on external circumstances (gathering darkness) and partly on a tendency to imitate others who have already performed the action in question. In this case the proportion y(t) of people who have performed the action can be described24 by the equation dy/dt = (1 y)[x(t) + by], (i) where x(t) measures the external stimulus and b is the imitation coefficient. (a) Observe that Eq. (i) is a Riccati equation and that y1(t) = 1 is one solution. Use the transformation suggested in 33, and find the linear equation satisfied by v(t). (b) Find v(t) in the case that x(t) = at, where a is a constant. Leave your answer in the form of an integral.