MATH vertical direction

							     MATH230                                                 Differential Equations
                                         Spring 2008
                                      Homework assignment 10
                                       Due Tuesday, 04/08/08

     (1) In this exercise we will consider a system of differential equations similar to the one we
         have been working with in class:
          dR           R
               3R(1 ) 1.4RF
          dt           6
          dF
               F  0.8RF
          dt

        (a) What is different about this system, as compared to the one discussed in class? What
            does this change mean in terms of the animal populations?
          Do you think that this change should affect the behavior of the system? Why or why
            not?




        (b) Do a phase-plane Analysis of this system. That means: Plot, by hand, all points at
            which the solution viewed in the phase-plane will move in a horizontal direction, and
            then again, and in a different color, all those points at which the solution will move
            in a vertical direction, when viewed in the phase-plane. Then decide which of the
            horizontal vectors will point right/left, and which of the vertical vectors will point
            up/down. This analysis will allow you to show the general direction a solution will
            be moving once it is in any of the four regions determined by the nullclines, (first
            quadrant.) Indicate those directions. (See regions A,B,C,D in Peter's Notes.)
(c) Finally compare the completed picture for this system to the one from class (which
    you may also have to complete at this point) next to each other. Do they indicate
    different or similar behavior of solution curves? Draw the two pictures and
    comment.




(d) Now look at some solution curves for both systems using the DEE applet. Do the
    solutions behave differently? Describe. Explain if you can, or express your surprise
    if you cannot.
 (2) Given are eight systems of differential equations in x and y, and four "direction fields" (i.e.
     phase-planes with direction vectors...) For each of the direction fields, determine which of
     the systems generated it, and explain how you know that your choice is correct based on
     ideas from the phase plane analysis we did in class (nullclines, specific features or directions
     at certain points, etc.) You should do this without using technology.

           dx                   dx                     dx                     dx
               x                  x 2 1                x  2y                2x
       (a) dt               (b) dt                 (c) dt                 (d) dt
           dy                   dy                     dy                     dy
               y 1               y                      y                   y
           dt                   dt                     dt                     dt


           dx                   dx                     dx                     dx
            x                  x 1                x 2 1              x  2y
       (e) dt               (f) dt                 (g) dt                 (h) dt
           dy                   dy                     dy                     dy
               2y                  y                    y                    y
           dt                   dt                     dt                     dt

 (I)                                                  (II)
                                                               




 (III)                                                (IV)
(3a) Below you see the graph of the x(t) and y(t) graphs of a solution curve to a system of
    two differential equations in x and y. (We have initial condition x(0)=2 and y(0)=-2.)
    Sketch the corresponding solution curve in the phase plane as accurately as you can, and
    indicate the direction in which the solution moves in the phase plane.




 (b) Now in reverse: Given is the direction field and three solution curves viewed in the
    phase plane for some system of two differential equations in x and y. Make a qualitative
    sketch of the x(t) and y(t) graphs corresponding to each of the solution curves. Precision
    is not as important here, but qualitatively everything should be correct. (You may want
    to use the next page, where I also printed another copy of this...)