306X Homework #6 due beginning of class, Tuesday, October 22, 2002.

Problems are from Blanchard, Devaney, and Hall, 2nd ed., unless otherwise noted.

[3.3]
  2.    (Phase portrait for problem 3.2.2.)

[3.4] Complex eigenvalues.
  3.    EXCEPT ... use the matrix -4 6 instead.
                                  -6 2 
        For part (c), do a rough hand-drawn phase portrait.
        Drawing nullclines will help the accuracy of your picture.
        Aslo, hand-sketch the x vs. t graph for one of the solution
        curves in your portrait.
  9.    EXCEPT ... use the matrix -4 6 instead.
                                  -6 2
	Use Maple to draw and superimpose the direction field and
	the solution of the IVP.
        Recall that you can use "display" to combine several plots:
           plot1:=plot(whatever):
           plot2:=plot(whatever):
           with(plots):
           display(plot1,plot2);

  3.    EXCEPT ... use the matrix -2 4 instead.
                                  -2 2

X.  Use Maple's ability to do parametric plots

         plot([x,y,t=whatever..whatever]);

    to make the following picture: a smiley face, with circles for the 
    head and the eyes and a semicircle for the mouth.
    

[3.5]
	2. and 6. (Linear system with repeated eigenvalue.)
	          Skip 2(c). Do 2(d) by hand using eigendirections 
	          and nullclines. Do (e) by hand using portrait from (d).
		  Do 6(c) with Maple.
	18.       (Linear system with a 0 eigenvalue.)


[3.6]
	34. (Damping for "most rapid approach to equilibrium".)
	    Remember that the ultimate approach to equilibrium 
            is governed by the slowest-decaying term in the solution:
	    the term corresponding to the less negative eigenvalue.