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Phase Diagrams [ Used for Autonomous D.E. with form dy/dx = f(y) ]

In a Nut Shell: First order differential equations of the form

dy/dx = f(y)

where the function on the right hand side does not depend on the independent variable, x,

are called "autonomous" and are amenable to separation of variables. Solutions for y(x)

may be quite complicated depending on the expression for f(y).

Critical points, y_o, used to identify stability occur where f(y_o) = 0. Plots on the real

axis line, y, showing the critical points along with the slopes between the critical points

are termed phase diagrams. They are useful in helping construct slope fields used to

identify where the response, y(x), is stable, unstable, or semi-stable.

Strategy for construction of the Phase Diagram

Step 1	Find the critical points. i.e. where the slope is zero. i.e. f(y) = 0
Step 2	Construct a real axis, y, and mark the values of y where f(y) = 0
Step 3	Find the slopes on either side of the critical points.
Step 4	At each critical point, draw arrows indicating increasing values of slope, i.e. For positive slope use an arrow pointing in the + y direction. i.e. For negative slope use an arrow pointing in the ˗ y direction. If the arrows at the critical point are towards each other, then the critical point is stable (i.e.sink). If the arrows point at the critical point are away from each other, then the critical point is unstable (source) If the arrows are both in the same direction, then the critical point is semi-stable.

The autonomous differential equation: dy/dx = y² ( y² ˗ 4 )

Has critical points at y = 0 and y = ± 2 . i.e. f(y) = 0

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