# MATH vertical direction by sanmelody

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```									     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

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...)

```
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