For the autonomous differential equation dy/dt = y(4 - y), find the equilibrium solutions and determine whether each equilibrium is stable or unstable.

Step 1: Find the equilibrium solutions.

Equilibrium solutions occur where dy/dt = 0.

Set the right side equal to zero:

y(4 - y) = 0

Step 2: Solve for y.

Using the zero-product property:

y = 0
or
4 - y = 0

Therefore:
y = 0 or y = 4

These are the equilibrium solutions.

Step 3: Analyze the sign of dy/dt below y = 0.

Choose a test value such as y = -1:

y(4 - y) = (-1)(5) = -5

Therefore, dy/dt < 0 when y < 0. Solutions move toward smaller y-values.

Step 4: Analyze the interval 0 < y < 4.

Choose y = 2:

y(4 - y) = 2(2) = 4

Therefore, dy/dt > 0. Solutions increase, so they move away from 0 and toward 4.

Step 5: Analyze the interval y > 4.

Choose y = 5:

y(4 - y) = 5(-1) = -5

Therefore, dy/dt < 0. Solutions decrease, so they move toward 4.

Step 6: Determine the stability of y = 0.

Immediately below 0, solutions move downward, away from 0.
Immediately above 0, solutions move upward, also away from 0.

Therefore, y = 0 is an unstable equilibrium.

Step 7: Determine the stability of y = 4.

Below 4, solutions increase toward 4.
Above 4, solutions decrease toward 4.

Solutions on both sides move toward y = 4.

Therefore, y = 4 is a stable equilibrium.

Thus, the equilibrium solutions are y = 0 and y = 4. The equilibrium y = 0 is unstable, while y = 4 is stable.