The differential equation has equilibrium solutions at ,

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

Problem

The differential equation

\dfrac{dy}{dt}=y^{3}-4y

has equilibrium solutions at

y=-2

y=0

, and

y=2

. Which statement most accurately describes the stability of these equilibria and the long-term behavior of a solution satisfying

y(0)=1

? (a)

y=-2

and

y=0

are unstable,

y=2

is stable, and the solution with

y(0)=1

approaches

y=2

t\to\infty

. (b)

y=-2

and

y=2

are unstable,

y=0

is stable, and the solution with

y(0)=1

approaches

y=0

t\to\infty

. (c)

y=-2

and

y=2

are stable,

y=0

is unstable, and the solution with

y(0)=1

approaches

y=2

t\to\infty

. (d)

y=-2

is stable,

y=0

and

y=2

are unstable, and the solution with

y(0)=1

approaches

y=-2

t\to\infty