MathHelp

Autonomous Differential Equations i.e. D.E. of the form dy/dt = f(y)

Example: Find the critical points then sketch the phase diagram for the following

differential equation:

dy/dt = (1/2) y ( y ˗ 2 )² ( y ˗ 4 )

Strategy: Step 1: Find the critical points by setting f(y) = 0.

Critical Points: f(y) = 0 at y = 0, y = 2, and y = 4

Step 2: Identify each region where slopes may change from positive to negative, stay the

same, or change from negative to positive. Evaluate the slopes on each side of the critical

points by taking values within each region.

The four regions in this example include: y < 0, 0 < y < 2, 2 < y < 4, y > 4

i.e. For y = ˗ 1, dy/dt = 45/2 > 0, for y = 1, dy/dt = ˗ 3/2 < 0

For y = 3, dy/dt = ˗ 3/2 < 0, and for y = 5, dy/dt = 45/2 > 0

Step 3: Plot Phase Diagram:

Note: A right directed arrow indicates that the dependent variable, y, is increasing.

A left directed arrow indicates that the dependent variable, y, is decreasing.

Results: At the critical point y = 0, the response is stable.

At the critical point y = 2, the response is semi-stable.

At the critical point y = 4, the response is unstable.

i.e. In the neighborhood of y = 0, the value of y(t) tends toward a stable value.

In the neighborhood of y = 2, the value of y(t) hovers around a stable value.

In the neighborhood of y = 4, the value of y(t) departs from a stable value.

Return to Differential Equations Notes