The differential equation has equilibrium solutions at and

MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations

Problem

The differential equation

\dfrac{dy}{dx}=(4-y)(y+1)

has equilibrium solutions at

y=-1

and

y=4

. Two solution curves satisfy

y(0)=0

and

y(0)=3

, respectively. Which of the following statements best compares the long-term behavior of these two solutions as

x

increases? (a) Both solutions decrease and approach

y=-1

x\to\infty

. (b) Both solutions increase without bound as

x\to\infty

. (c) Both solutions increase and approach

y=4

x\to\infty

. (d) The solution with

y(0)=0

approaches

y=-1

, while the solution with

y(0)=3

approaches

y=4