# 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):

See attached file for more information.

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

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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 ...

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