Question

Solve the differential equation

dx/dt =x²(1−x), And find the critical points. Identify the stability of the critical points.

Answer 1

To solve the differential equation dx/dt = x²(1−x), we separate variables and integrate. We find the critical points by setting dx/dt = 0 and solve for x. To determine stability, we analyze the sign of dx/dt around each critical point.

Step-by-step explanation:

To solve the differential equation dx/dt = x²(1−x), we can separate the variables and integrate both sides. Rearranging the equation gives dx/(x²(1−x)) = dt. Now we integrate both sides. The integral of dx/(x²(1−x)) can be solved using partial fraction decomposition. After solving for x, we can find the critical points by setting dx/dt = 0 and solving for x. To determine the stability of the critical points, we can analyze the sign of dx/dt around each critical point. If dx/dt is positive, the critical point is unstable. If dx/dt is negative, the critical point is stable.

Answer 2

In summary, the critical point x = 0 is unstable, x = 1 is stable, and x = -1 is unstable.