Consider the logistic differential equation

MATHEMATICS · Subjective · AP · Calculus BC · Differential Equations · Differential Equations

Problem

\dfrac{dy}{dt}=k\,y\,(a-y)

, where

k

and

a

are positive constants. Explain whether the population grows fastest when

y

is small, medium, or large relative to the carrying capacity

a

, and give a brief reason based on the expression for

\dfrac{dy}{dt}