A family of slope fields is generated by the differential equation

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

Problem

\dfrac{dy}{dx}=y^{2}-c

, where

c

is a positive constant. Students compare slope fields for

c=1

c=4

, and

c=9

. Which statement best describes how the positions and stability of equilibrium solutions change as

c

increases? (a) The two equilibrium lines move closer to the

x

-axis, but their stability types remain unchanged. (b) The two equilibrium lines move farther from the

x

-axis, and both become stable. (c) The two equilibrium lines move farther from the

x

-axis, but their stability types remain unchanged. (d) The lower equilibrium line disappears for

c>1

, leaving only a single equilibrium line above the

x

-axis.