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\centerline{Mathematics 304, section 2 -- Ordinary Differential Equations}
\centerline{Discussion 2 -- Phase Portraits of $2 \times 2$ systems $X' = AX$}
\centerline{September 29, 2004}
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{\it Goals}
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Today we want to work through a number of examples of $2 \times 2$
linear systems of ODE of the form
$$X'(t) = \pmatrix{a&b\cr c&d\cr}X(t)$$
and understand the overall behavior of the set of solutions.  
The goal is to have each group work out one example and present your
results to the whole class on the blackboard by the end of the 
class period.  Then we can discuss the properties of all these examples.
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{\it Discussion Question}
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\noindent
For your given system:
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\item{1)}  Determine the eigenvalues and eigenvectors for your
matrix $A$.
\item{2)}  Write down the general solution 
$$X(t) = c_1 e^{\lambda_1 t}V_1 + c_2 e^{\lambda_2 t}V_2$$
\item{3)}  Sketch the straight-line solutions (that is, solutions
with $c_1 = 0$ or $c_2 = 0$, indicating the direction of increasing
$t$.  Then,
\item{4)}  In a qualitative way, sketch the behavior of other
solutions (that is, solutions where $X(0)$ is not an eigenvector), giving
the ``phase portrait'' of the system.
\item{5)}  Describe the overall behavior of solutions of your system.
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{\it Your Systems}
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\item{} Group 1 (Alison, Elizabeth C., Rebecca) -- the system
$$X' = \pmatrix{-3 & 2\cr -1 & 0}X$$
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\item{} Group 2 (Nicole, Ben, Meghan) -- the system
$$X' = \pmatrix{0 & 2\cr -1 & 3}X$$
\item{} Group 3 (Ingrid, Mackenzie, Trevor) -- the system
$$X' = \pmatrix{-8 & 10\cr -5 & 7}X$$
\item{} Group 4 (Molly, Bridget, Caitlin, Elizabeth V.)  -- the system
$$X' = \pmatrix{8 & -10\cr 5 & -7}X$$
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{\it Assignment}
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Each group is also responsible for a ``clean copy'' of your
example, which will be distributed to the rest of the class
as part of the notes for this day's class meeting.  These are
due on Monday, October 4.
\bye