The differential equation models the rate of change of a population (in hundreds)…
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
The differential equation dxdy=y(1−5y) models the rate of change of a population y (in hundreds) with respect to time x (in months). No slope field is provided.
(a) Identify all equilibrium solutions and classify each as stable or unstable. Explain your reasoning.
(b) Describe qualitatively what the slope field would look like in the vicinity of the equilibrium solutions. Your description should reference the sign of dxdy in each region between equilibria.
(c) Suppose y(0)=0.5. Without performing any numerical calculations, use your answer from part (b) to describe the behavior of the solution as x increases. In particular, indicate any long-term limiting value.
(d) Interpret the limiting value you found in part (c) in terms of the context of the population model.