Equilibrium solutions A differential equation of the form y′(t) = F(y) is said to be autonomous (the function F depends only on y). The constant function y = y0is an equilibrium solution of the equation provided F(y0)= 0 (because then y′(t)=0, and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal line segments in the direction field. Note also that for autonomous equations, the direction field is independent of t. Consider the following equations.

a. Find all equilibrium solutions.

b. Sketch the direction field on either side of the equilibrium solutions for t ≥ 0.

c. Sketch the solution curve that corresponds to the initial condition y(0) = 1.

y′(t) = y(y −3)(y + 2)

Problem 47E

Equilibrium solutions A differential equation of the form y′(t) = F(y) is said to be autonomous (the function F depends only on y). The constant function y = y0is an equilibrium solution of the equation provided (because then y′(t)=0, and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal line segments in the direction field. Note also that for autonomous equations, the direction field is independent of t. Consider the following equations.

a. Find all equilibrium solutions.

b. Sketch the direction field on either side of the equilibrium solutions for t ≥ 0.

c. Sketch the solution curve that corresponds to the initial condition y(0) = 1.

y′(t) = y(y −3)(y + 2)

Step 1</p>

In this problem we have to find all the equilibrium solutions of

a. Find all equilibrium solutions.

We get the equilibrium solution by setting

Thus are the equilibrium solutions.

Step 2</p>

b. Sketch the direction field on either side of the equilibrium solutions for t ≥ 0.

The following figure shows the direction field on either side of the equilibrium solutions for t ≥ 0.

We know that

An equilibrium solution is stable if nearby solutions are attracted to it.An equilibrium solution is unstable if nearby solutions are repelled from it.In the figure we see that, is a stable equilibrium solution and are unstable equilibrium solutions.

Step 3</p>

c. Sketch the solution curve that corresponds to the initial condition y(0) = 1,

We have

Integrate on both sides, we get

(where c is integration constant)… (1)