Math 293, Homework 6,    due Wed. October 9, 1996


1) Make up a word problem, set up the equations, and solve
   the equations using MATLAB and ODE23, plot your
   solution.  The  equations should be a pair of 
   coupled first order differential equations.

   Sentences like "assume x changes at a rate proportional
   to y squared with a proportionality constant of 6" do
   NOT count really as word problems!!
   Try to think of a model of some physical or social 
   situation where you think the model is kind of 
   reasonable.

   You may also check your work using pplane or using 
   Bob Terrel's Applet 'de'. For help with ODE23, for a copy
   of pplane, or for info on 'de', consult the course
   WWW page.

   Hand in: words, ODEs and ICs, MATLAB work, MATLAB output,
   your interpretion of the result.

2) You should be able to do 4.1.1-20 and 4.1.26
   Hand in 4.21a.
   Also do 4.21a) by the following method: find the 
   derivative (with respect to time) of
   x^2 + y^2 assuming only that x and y solve the given system,
   i.e., without using the solution to the ODEs for x(t) and y(t). 
   Show how this result implies that the trajectories must be
   circles.   Moral: sometimes you can find the phase plane
   trajectory without  finding  the time history of the motion.

*) You should be able to do 5.1.1-6 and 5.1.11-20
             
*) For  A = [ 2  3 ] a two by two matrix, and [b] = [1], a 2x1 col vector,
            [ 1  4 ]                                [5]

  a) Find Ab, 
  b) the transpose of A, 
  c) the diagonal part of A, 
  d) write the equation Ax=b in long hand (as scalar equations),
  e) solve Ax=b for x.
  f) check your solution.
  g) Calculate  AC   where C = [ 2  0 ].  Note that this just multiplies the
                               [ 0  3 ]   columns of A by 2 and 3.

3) For A = 1 2 3 4 5   (5x5 matrix)  and    b = 1   (5x1 col vector)
           5 4 3 2 1                            2
           1 2 0 0 0                            3
           5 5 4 4 4                            4
           3 0 3 0 3                            5

   repeat steps a-f from problem 2.  You may use MATLAB if you like
   (See WWW page for a script file with a list of useful matrix matlab
    commands).