Math 331, Fall 2002: Problems 13-16

**NOTE:** *Each exercise is worth 10 points and can be
turned in at any time before its ``expiration date''.
At the end of the semester, I will expect you to have
turned in at least 2/5 of the exercises assigned. If you do more, I
will pick your best grades. If you do less, the missing grades will be
counted as zeros. Altogether, these will count the same as one project.
*

**13.**- (
*expires 10/20*) Find all the solutions to the differential equation

Among them, single out the one for which . [*Hint: read the help page for*`dsolve`*, or just do it in your head. It is that easy.*] **14.**- (
*expires 10/20*) Have`Maple`find analytic solutions to the following system of differential equations,

with initial conditions: . Let us denote the solutions by (since they depend on the parameter ).For taking all integer values from -10 to 10, and , plot the functions in blue, and the functions in red, all on the same graph. (Yes, you will then have 42 functions plotted on the same graph.) [

*This is certainly a case when you don't want to retype the functions that*`Maple`finds. You will almost certainly need to read the help page for`dsolve`*. I also found*`subs`*,*`unapply`*, and*`seq`*useful.*] **15.**- (
*expires 10/20*) For the functions and found in problem #14, plot the parametric curves for integer values of between and and on the same graph. Use the`view`option of plot to only show what lies in the region , and use a sequence of colors so that each solution is a different color. [*HINT: you might find something like*`seq(COLOR(HUE,i/11),i=0..10)`*useful for the latter.*] **16.**- (
*expires 10/20*) Find all the fixed points of the system

a fixed point being a solution for which*both and*stay constant. For each of these points, describe the behavior of the solutions that have initial conditions nearby. You can use`Maple`to figure out what happens for nearby points, or you can use more mathematical methods.

**NOTE:** *The fact that there are various notations for
differential equations is purely intentional.*