Qualitative Analysis: Answer to Example3

Example: Consider the first order system

1.

Graph the nullclines for this system.

2.

Insert direction field "arrows" at the nullclines

3.

Determine the general direction of the direction field in each of the regions cut out by the nullclines.

4.

Sketch possible phase portraits for the two initial conditions indicated by "dots" for as long as the phase portraits stay within the graph's frame.

5.

Find all the equilibrium points of the system.

Answer:

1.

The x-nullclines can be obtained by setting x'=0, i.e.

The y-nullclines can be obtained by setting y'=0, i.e.

For 2, 3, 4, use the following graph:

5.

The equilibrium points of the system are the intersection points of the x-nullclines and y-nullclines. Therefore, we get

,

which yields x=2 and . Hence, we have two equilibrium points

.

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