The family of autonomous differential equations

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

Problem

\frac{dy}{dx} = y^2 - n\,y

, where

n

is a real parameter, produces different slope fields as

n

varies. (a) Show that the number of equilibrium solutions depends on the sign of

n

. List the equilibrium values of

y

for

n < 0

n = 0

, and

n > 0

. (b) For

n = 2

, sketch the slope field in the rectangle

-2 \le x \le 2

-1 \le y \le 3

, paying particular attention to the slopes along the lines

y = 0

and

y = 2

. (c) For

n = -1

, repeat the sketching task described in part (b) on a separate diagram. (d) Compare the long-term behaviour of a solution with initial condition

y(0) = 1

for

n = 2

and for

n = -1

. Use both your slope fields and analytical reasoning based on the differential equation

\frac{dy}{dx} = y^2 - n\,y

to justify your conclusions.