COALITION MATH 152, SPRING 1998

              (S. A. Fulling, assisted by Vera Rice)

                             Day 24.T


    1.  The usual paper shuffling:  Old homework, CAPA, revised 
    new homework, TWO lab sheets (Lab 25 recommends consulting 
    Lab 26 for examples of plot commands).

    2.  Announce test statistics and curve and comment on two 
    hard problems (moment of inertia and integral of ln x).

    3.  Second-order homogeneous linear differential equations

        A.  For purposes of a class exercise, construct your 
        personal ODE problem from your ACS username:  Subtract
        from each digit, dj, to get 4 numbers, (a1,a2,a3,a4).
        Your equation and initial data are

        y" + a1 y' + a2 y = 0,   y(0) = a3, y'(0) = a4.

        (I construct my ODE as an example.)

        B.  Solve your equation on paper to turn in Thursday.
        (Some people with instructive equations will also get 
        plastic.)  Check you teammates' work before Thursday!
        -- For two reasons:

            i.  experience with several equations

           ii.  mercy on the grader

        C.  I solve my equation as an example (it is Case vi
        below).

        D.  I present plastic with the problems set up to the 
        persons prechosen to demonstrate various cases.  (The 
        parenthetical characterization is to help the teacher 
        find appropriate student numbers, not to be told to the 
        students.)

            i.  Simple exponential (d1 = 4, d2 < 4).
            Interrupt to introduce the functions sinh and cosh 
            and their use in solving such initial value problems.
                
           ii.  Simple trig (d1 = 4, d2 > 4)
            Interrupt to introduce the basic facts about complex 
            numbers.
I FOUND THAT STUDENTS WERE NOT PREPARED TO DO THESE PROBLEMS 
WITHOUT PROMPTING, SO I HAD TO GIVE THE COMPLEX NUMBER 
MINILECTURE BEFORE WAITING FOR THE FIRST TEAM TO PRESENT CASE i.

                a.  i^2 = -1 and the usual rules of algebra 
                apply.  (The two square roots of -1 are i and
                - i.)

                b.  e^{i \theta} = cos \theta + i sin \theta
                and e^z obeys all the usual identities for the 
                exponential function, even when z = x + iy is 
                complex.

                c.  Complex conjugate:  (x + iy)* = x - iy
                (when x and y are real).
    
          iii.  Perfect square (nonzero double root) 
                d1 = 8, d2 = 8  or  d1 = 6, d2 = 5  or
                d1 = 0, d2 = 8  or  d1 = 2, d2 = 5
THIS TEAM DID NOT DO THE PROBLEM CORRECTLY WITHOUT HELP; THEY 
WERE SENT BACK TO CHECK THE CORRECT ANSWER FOR NEXT TIME.

           iv.  Zero root (d2 = 4; double if d1 = 4 also)
I HAD ONE STUDENT IN THE CLASS WITH A DOUBLE ZERO, AND ONE WITH A 
NONZERO DOUBLE ROOT.  IT WAS NECESSARY TO SWAP THE LATTER'S 
NUMBER WITH ANOTHER TEAM TO GET A FAIR DISTRIBUTION OF CLASS 
PARTICIPATION OPPORTUNITIES.

AT THIS POINT THE TIME RAN OUT.
            v.  Damped oscillations 
                (d1 > 4,  d2 > 4 + (d1 - 4)^2/4 )
                Stop here if out of time.

           vi.  Overdamped system
                (d1 > 4,  d2 < 4 + (d1 - 4)^2/4 )

          vii.  Growing oscillations
                (d1 < 4,  d2 > 4 + (d1 - 4)^2/4 )
    
         viii.  Runaway growth
                (d1 < 4,  d2 < 4 + (d1 - 4)^2/4 )