You are encouraged to hand in joint work. Also, feel free to bug the
prof or TA about computer problems. Also remember your minor project
is coming up! You may find the material under systems of equation on netmath
more helpful than the program you have been using.

Notation: In this problem set, [ a b; c d] is a matrix whose first row is (a, b)
and whose second row is (c, d).  That's how you enter matrices in many
computer packages such as MATLAB.  Likewise, [x ; y] is a 2x1 matrix (that is,
a column vector) whose upper entry is x and whose lower entry is y.

In the following problems, you are asked to make a direction field plot
of the solutions of the vector equation given by
dX(t)/dt = A X(t).
Here A is one of the listed matrices below, and X(t) is the vector [x(t) ; y(t)].

Also, mark the straight-line solutions you see. What happens to
solutions as t --> infinity?
 

1. A =  [ -2 1 ; 3 -2 ].

Remember to mark the straight lines and to tell what the solutions do as
t--> infinity.

2. A =  [2 1; 3 2]. (Same reminder).
 

3. A = [ 2  2; 2 -2]. (Same reminder)

4. A = [-1 -2; 2 -1] ( ditto).

5. A = [ -1 -2; 2 1] ( ditto).
 

Explanation: Here is an extra credit exercise to give you
a hint of what is going on (you can, of course, always look at the book!)

6* A very clever student conjectures that if you take the matrix
A=[a b; c d]  and set the determinant of [a-x b; c d-x] equal to zero, the roots
of this equation determine what sort of picture you will get. Carry this
out and hypothesize what roots correspond to what behavior. Conjecture!
Of course, you can read this in the book, but if you work it out yourself you
will realize that math is not at all mysterious! At the bottom of the page
for the systems of ordinary differential equations option in netmath, there
are a series of programs for computing "eigenvalues", which are these numbers,
and the vectors which go with them (the 3X3 example can be changed to do a
matrix of any size). Experiment with this as well.