306X Homework #4 due beginning of class, Thursday, October 3, 2002.

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

NOTE: This homework involves a significant amount of Maple use. 
Get started right away. Remember that you cannot expect instantaneous 
printouts of your Maple worksheets. 

2.1   2. (Find the equilibria of two 2D systems.)
	 In (ii) use a different DE: divide the whole RHS by 20.
	 Additionally:
	 (a)  Use Maple to make a direction-field picture for each system.
              Choose the scales to include all equilibria.
              


	 (b) Find the equations for the nullclines of each system,
              and sketch them on your direction fields.


[2.2]   8. In (c), use Maple to create the vector field and
           direction field pictures.
	   In (d), hand-sketch solution curves on your direction
           field hard-copy.
	13-16. (Matching phase portraits and time-curves.)
	   



[2.3]  19. (A "triangular" or "partially decoupled" system.) 
         Skip part (d).

[2.4]   2. as follows.
         (i) Do part (a).
         (ii) Use Maple to plot the verified solution for t between -3 and 1.
              Plot x and y together versus t, and y versus x.
	      On the latter plot, put a dot at Y(0).
              Here is how to do plotting in Maple.
              


         (iii) Use Euler's method with a single step of size of 0.5 to
              approximate Y(0.5), for the IVP given. How big are the
              errors in your approximations of x(0.5) and y(0.5)?

[2.5] 2. (Equilibria of the Lorenz system.)