MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
In a closed community of fixed size N, let y(t) represent the number of individuals who have heard a certain rumor at time t (in hours). Social scientists propose that the instantaneous rate at which new individuals hear the rumor is proportional to the product of the number who have heard it and the number who have not.
Assume 0<y(t)<N and that y(t) is differentiable for t≥0.
(a) Derive a differential equation for y(t) involving a single parameter k. Express your answer in terms of y and N only; do not solve for y(t).
(b) Verify algebraically that your differential equation from part (a) is autonomous (i.e., the right-hand side depends only on y and not explicitly on t). Explain the modeling significance of this property.
(c) Determine all equilibrium values of y and classify each as stable, unstable, or semistable purely from the form of the differential equation, discussing how different values of k (positive or negative) influence your classification.
(d) Suppose k is positive and y(0)=1. Provide a reasoned argument, without solving the equation, that y(t) cannot exceed N for any t>0, and explain how this reflects the constraints of the real-world situation.