COALITION MATH 152, SPRING 1998

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

                             Day 23.T


    1.  The usual paper shuffling:  Old homework, CAPA, revised 
    new homework, lab sheet if available.

    2.  Remarks on technical part of project proposal/report 
    (Point 4 of previous day).

    3.  First-order differential equations (accelerated to make 
    up a lost day)

        A.  Solve a (nonlinear) separable equation with initial 
        data.  (This would have been a team exercise if we had 
        time.)  y' = x^2/(1 + y^2),  y(2) = 1.

        B.  The integrated relation is hard to solve for y in 
        this example, so use it as an occasion to demonstrate:

            i.  Verification by implicit differentiation

           ii.  Maple commands DEplot1 and implicitplot
MINOR DISASTER:  DEplot1 IS A RELEASE 3 COMMAND -- NO LONGER 
EXISTS IN RELEASE 4.  ALSO, dfieldplot NO LONGER DOES WHAT IT DID 
IN RELEASE 3.  MY SYNTAX EXAMPLES NEED TO BE UPDATED (TO RELEASE 
5?)
        C.  Quickly mention other ODE-related Maple commands:
        dfieldplot, dsolve, plot, display

        D.  Linear equations

            i.  Homogeneous case
                
                a.  Separable!  Solve as discussed above.

                b.  This is not the same meaning of "homogeneous" 
                as on pp. 972-974 of Stewart 3.  That meaning is 
                LESS important than this one.  ("Homogeneous" 
                also refers to materials with no spatial 
                dependence -- still another meaning.)

           ii.  Nonhomogeneous case, such as
    
                y' + (2/x) y = (cos x)/x^2 

                a.  Solve the corresponding homogeneous equation.

                    y_h = x^{-2}  in example.

                b.  Multiply equation by integrating factor 

                    mu = 1/y_h 

                c.  By magic, the left side is d(something)/dx
                (by the product rule in reverse).

                     d/dx (x^2 y) = cos x  in the example.

                d.  Evaluate the antiderivative and solve for y.

                    y = (sin x)/x^2 + C/x^2  in example.

                The term with the arbitrary constant is the most 
                general solution of the homogeneous equation.

        E.  If time permits, show prepared transparencies with 
        example of Euler's method.  Emphasize:
    
            1.  Common error of miscounting the steps.

            2.  Euler is NOT the method of choice for solving 
            ODEs numerically.  More complicated methods give 
            greater accuracy for the same computation time.

            Euler  is to  Runge-Kutta or Predictor-Corrector
            as Riemann sums  are to  Simpson's rule.

ALL THIS WAS ACCOMPLISHED WITH ONLY A BIT OF RUSH.