Consider the one-parameter family of differential equations dy dx = k ,y ,(10-y)

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Consider the one-parameter family of differential equations

\frac{dy}{dx} = k\,y\,(10-y)

, where

k > 0

. (a) Determine all equilibrium solutions in terms of

k

. (b) For

k = 0.4

, calculate the slope at the point

(0,2)

. (c) For

k = 1.2

, find the ratio of the slope at

(0,2)

to the slope you obtained in part (b). (d) State whether changing

k

alters the location of the equilibrium solutions, and justify your answer by referring to the differential equation.