Math 293, Homework 8 
       due in lecture on Wednesday October 23, 1996.

(*) = don't hand in

*) You should be able to do problems of the form 5.2.1-37

1) 5.2.13. Pick and state some initial condition and find
   c1 and c2 to satisfy that initial condition.


2) Let x(t) = [ x1(t) x2(t) x3(t) x4(t) ]'  where []' is the transpose of [].
       A    = [ -10    -2     3    -4;
                  5    -6     7    -8;
                  1    -2     3     5;
                  e    -pi    1    -1    ] ;

          ( where pi=3.1415... and  ln(e) = 1 )

   You are given that x satisfies the system of ODEs  x'   = A x 
                      and the initial condition       x(0) = [ 0 0 1 -3  ]'       

   a) Find x(2) using the eigenvalue and eigenvector method.
      That is find the 4 numbers x1(2), x2(2), x3(2), and x4(2).
      You may assume, correctly, that A has 4 eigenvectors.
      
      (The course WWW page has enough samples to get you through
       the needed MATLAB commands.  The text and lecture indicate
       the recipe that is to be applied.)
       
      Hand in a tidy script file, not a rambling interactive 
      session of commands.)

   b) Find x(2) using ODE23 or ODE45.

(If any of your 4 answers  from parts (a) and (b) differ
         by more than a percent you have made a mistake.)

   c) Optional challenge 1: 
      How close can you get the answers of (a) and (b)?
      At that accuracy which takes less computer time to find?

   d) Optional challenge 2:
      How compact can you make your solutions?