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Separation of Variables, System Progress, and Phase Diagrams

(1) Solve by separation of variables and solution to the ODE (see attached file for formula).
(2) Solve the equation (see attached file for formula).
(3) For the given graph, re-sketch a graph for x01 and indicate the system progress from x01 as t --> infinity. Identify any critical points as stable, unstable, and semi-stable (see attached file for formula).
(4) Draw and label the phase diagram. Indicate the system progress (using arrows and explain. Identify any critical points at stable, unstable, or semi-stable: (see attached file for formula).
(5) Draw and label the phase diagram. Indicate the system progress (using arrows and explain. Identify any critical points at stable, unstable, or semi-stable: (see attached file for formula).
(6) A harvesting model is represented (by the formula in the attached file):

See attached file for more information.

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The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.

1.
The equation is:
(1.1)
Then:

(1.2)
Applying initial condition:

(1.3)

Therefore:
(1.4)
The solutions look like:

2.
The equation is:
(2.1)
This is a separable equations:

(2.2)
At t=0 we have therefore:

(2.3)

As we see the solution approaches
(2.4)
Note that at this limit
(2.5)
That is, is a stationary point of the equation.
At that point:
(2.6)
The first derivative at the equilibrium point is negative, hence the equilibrium point is stable.

3.

The phase diagram is:

There are two equilibrium points and corresponding to the situation
• Point If then and decreases - moving away from
• Point If then and decreases - moving towards and away from
• Point If then and increases - moving away from
We see that is a semi-stable point. If we start slightly to its ...

Solution Summary

The solution discusses the separation of variables, system progress and phase diagrams.

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