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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 (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...)