Consider the differential equation with equilibrium solutions at and

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

Problem

Consider the differential equation

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

with equilibrium solutions at

y=0

and

y=1

. Which statement most accurately describes the behavior of the solution curve that passes through the point

(0,2)

x

increases? (a) The curve decreases and crosses

y=1

since the slope there is negative, continuing downward afterward. (b) The curve decreases and approaches

y=0

, leveling off without crossing that line. (c) The curve decreases and approaches

y=1

, leveling off without crossing that line. (d) The curve increases without bound because

\dfrac{dy}{dx}

is always positive when

y>0