# 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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1.

The equation is:

(1.1)

Then:

(1.2)

Applying initial condition:

(1.3)

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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)

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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.

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3.

The phase diagram is:

There are two equilibrium points and corresponding to the situation

â€¢ Point If then and decreases - moving away from

â€¢ Point If then and decreases - moving towards and away from

â€¢ 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.