306G        Study Guide Test #3      

 

Problem 1: (Section 3.6)

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

 

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

 

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.

 

(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.

 

 

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.

 

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.

 

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).