New Homework Set 8:

Still do Maple problem on damping.
See: http://galileo.math.siu.edu/Courses/305/S99/dd.html

Same as before:
5.1: 1, 3, 6, 8, 14, 16, 18, 20, 22
5.2: 1, 4, 9, 14, 15

For 5.3 do the following:

1. For each differential equation below find the first 5 terms of the 
Taylor series of the solution. Do this by hand, showing all YOUR  work.
           2
a. y'' +  x  y' + y = 0,   y(0)=1,  y'(0)=0.

    2
b. x y'' + (x+1)y' + 3(ln x)y = 0,   y(1)=2,  y'(1)=0

       4                4 
c. (1+x )y'' + y' + (1+x )y = 0,   y(2)=1,  y'(2)=1


2. Find a lower bound for the radius of convergence for the series in (a), (b)
and (c). Hint: For (c) you can use the maple command "solve(1+x^4);" to get
the complex roots of 1+x^4.

3. Use Maple to find the first 10 terms of the Taylor series solutions to (a),
(b) and (c). Your name must be printed on YOUR output.

Example:
> with(DEtools):
> Order:=10;
> dsolve({diff(y(x),x$2)+x*diff(y(x),x)+y(x)=0,y(2)=3,D(y)(2)=1},y(x),series);

                              2              3               4
y(x) = 3 + x - 2 - 5/2 (x - 2)  + 4/3 (x - 2)  - 1/24 (x - 2)  - 1/4

           5   13         6                7    37         8    71
    (x - 2)  + --- (x - 2)  + 5/504 (x - 2)  - ---- (x - 2)  + -----
               144                             2688            36288

           9            10
    (x - 2)  + O((x - 2)  )

4. Notice that the series for (a) is alternating and that the terms are clearly
decreasing. This allows us to estimate the accuracy of the partial sums. If
the true sum is S and the n-th partial some is Sn, then it is not hard to show
that |S-Sn| <= An. How many terms of the series for (a) are needed if we want
to have an accuracy of .01 on the interval [-1,1]?