COALITION MATH 152, SPRING 1998

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

                             Day 22.R


    1.  Finish the Pac-Man exercise.
THIS TOOK THE ENTIRE PERIOD!  ALL OF POINT 5 WILL BE JUNKED AND 
REPLACED WITH A MORE STREAMLINED APPROACH NEXT DAY.  SOME OF THE 
MATERIAL WILL BE WITH THE TRANSPARENCY COPIES STUDENTS CAN 
PURCHASE.

    2.  Remarks on terminology (point 4 of previous day).
I WORKED THIS INTO THE REVIEW OF THE PROBLEM WHEN STARTING THE 
EXERCISE.

    3.  Review sessions and tests just like last time (except 
    possibly chem <-> phys reviews).  Math review 7:30 Mon.
    Math test 8:25 Thu.
    
    4.  Project proposal calls for "a technical analysis, based 
    on physical principles, of the functioning of your design."  
    What does this mean?

        A.  Depending on your design, you probably will hit the 
        ball with something.  What happens then?  Apply 
        conservation law....

        B.  Where does the ball go?  Remember Maple lab on flight 
        of a baseball, and many physics and math homework 
        problems.  

        C.  Do not design a device to send the ball through the 
        roof.

        D.  Later Maple labs will help analyze the effects of 
        drag.

        E.  The best targeting algorithm will be based on theory, 
        not just empirical calibration.

    5.  Introduction to differential equations
NEARLY ALL OF THIS HAD TO BE OMITTED AND LEFT TO THE READING.

        A.  Schedule for today and next Tuesday (easiest thing 
        first) 

            i.  Simple solvable equations (first-order separable) 
            (WILL BE ON TEST)

           ii.  Euler's method (lab Monday) 

          iii.  Direction fields (ii and iii will not be on test 
            but WILL be on a quiz or RAT Tuesday.)

           iv.  Nonhomogeneous linear first-order equations
     
        B.  A DE is to be solved for an unknown function, and 
        contains derivatives of that function.  We have seen 
        examples:

            i.  dy/dx = y' = ky => y(x) = C e^{kx}

           ii.  y' = f(x) (given) => y(x) = \int f(x) dx [+ C]

       C.  The general solution involves an unknown constant 
       until an INITIAL VALUE is given.  (Note:  C is not always 
       y(0).)

       D.  Other examples

           i.  y' = y^2  (nonlinear - defined later).  Solved 
           later.

          ii.  y'' + 4y = 0 (2nd order) => y(x) = A cos 2x + B 
          sin 2x.  (2 constants => need to know y(a), y'(a))

         iii.  y_1' = 2y_1 - y_2,  y_2' = y_1 + 2 y_2
           (system of DEs = eq. for vector-valued function y(x))

          iv.  \partial^2y/\partial x^2 = \partial^2y/\partial t^2
           (partial DE).  One solution is y(x,t) = sin(x - t) 
           (wave).

       E.  Separable first-order ODE:  y' = y^2.
           General structure:  y' = f(x) g(y)
           SEPARATE the variables:  dy/y^2 = dx
           Integrate; there is one arbitrary constant, not 2.
           Solve for y:  y = -1/(x + C).  Verify.

       F.  Observe:

           i.  Still one arbitrary constant, but it enters in a 
           nonlinear way.  C = - 1/y(0).

          ii.  What if y(0) = 0?  Then y(x) = 0.
               What if C = 0?  Then graph doesn't go through 0.

         iii.  When x = - C, solution "blows up" -- unlike linear 
           cases.

        G.  Example application (cf. CAPA).  #30, p. 512 
        (mixing).