Question

(a) Find all equilibrium solutions for the differential equation

dy/ dx = 0.5y(y-4) (2 + y).

(b) Draw some solution curves and determine whether each equilibrium solution is stable or unstable.

Answer 1

To find all equilibrium solutions, set the derivative equal to zero and solve for y. There are three possible solutions: y = -2, y = 0, and y = 4. To determine stability, analyze the sign of the derivative around each solution.

Step-by-step explanation:

To find all equilibrium solutions for the differential equation dy/dx = 0.5y(y-4)(2 + y), set dy/dx equal to zero and solve for y.

0 = 0.5y(y-4)(2 + y)

There are three possible equilibrium solutions: y = -2, y = 0, and y = 4.

To determine whether these equilibrium solutions are stable or unstable, we can analyze the sign of the derivative around each solution. If the derivative is positive on one side and negative on the other, the equilibrium solution is unstable. If the derivative is negative on both sides or positive on both sides, the equilibrium solution is stable.