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Solved: The propagation of a single action in a large population (for example, drivers
Chapter 2, Problem 27(choose chapter or problem)
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 described26 by the equation dy/dt = (1 y)( x(t) + by), (28) where x(t) measures the external stimulus and b is the imitation coefficient. a. Observe that equation (28) is a Riccati equation and that y1(t) = 1 is one solution. Use the transformation suggested in 25, 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.
Questions & Answers
QUESTION:
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 described26 by the equation dy/dt = (1 y)( x(t) + by), (28) where x(t) measures the external stimulus and b is the imitation coefficient. a. Observe that equation (28) is a Riccati equation and that y1(t) = 1 is one solution. Use the transformation suggested in 25, 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.
ANSWER:Step 1 of 5
(a)
Consider the given differential equation and initial condition is as follows:
The Riccati equation can be written as follows:
