**
**__306G
Study Guide Test #3 __

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__Problem 1__: (Section
3.6)

Find a function such that the trajectories
of the first order system lie on the level curves of .

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**See Example 1 on page 189
of sect. 3.6. Also see problems 10, 11 and 12 on page 193. **

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__Problem 2__: (Section
7.1 and 7.2)

Given an almost linear first
order system (S) of the form where are polynomials
in determine the following:

(a)-[ ]-the critical points
of (S), (b)-[ ]-the corresponding linear system near each critical point, (c)-[
]-the eigenvalues of each linear system.

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(d)-[ ]-Use the results of
(a)-(c) to classify the critical points of (S).

** **

**See Theorem 7.2.2 on page
487 and table 7.2.2 on page 488. See Example 3 on page 486 and Example 4 on
page 487.**

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__Problem 3__: (Sect.
7.3)

** **Do a
competing species problem like Example 1 one page 495 where coexistence of both
species is possible.

**See Example 1 on page 494.**

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__Problem 4__: (Sect.
7.3)

Do a competing species problem like Example 2 one page
497 where coexistence of both species is impossible.

**See Example 2 on page 497.**

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__Problem 5: (7.4 &
7.2)__

(a)-[ ]- Given the phase
portrait of a given non-linear system (S) of the form locate and classify the critical points
of (S).

(b)-[ ]- Given the phase
portrait of a given non-linear system (S) of the form identify the component plots of .

(c)-[ ]- Given the component
plots of for a given non-linear system (S) of the
form identify the phase portrait of (S).