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# Separation of Variables, System Progress, and Phase Diagrams

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(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):

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#### Solution Preview

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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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