i'd suggest first looking at
spring04-computer-projects/project-five-mathematica.nb

as an alternative, look at
spring04-computer-projects/project-five-old-stuff/project-four-mathematica.nb
this one may be generally edifying, too.

lastly, if you don't like either of those, then try
spring04-computer-projects/project-five-old-stuff/project-five-mathematica-fixed.nb



Date: Mon, 22 Mar 2004 17:49:30 -0600 (CST)
From: James Michael Rath <foo_address@ices.utexas.edu>
To: "a student" <someone@utexas.edu>
Subject: Re: project 5

> Our group is having some problems deciphering exactly what we are supposed to 
> use Maple for in this hw project. We used it to find the indicial qeuation and 
> the roots.

> so we were wondering if we 
> just had to solve everything by hand and then use Maple to find the terms, or 
> what do we need to do?

ok, you're on the right track.  computing the limits to find the 
coefficients in the indicial equation is a good use of maple.  so is using 
it to find the roots of the resulting quadratic.

there're further uses.  you're asked for the first six terms in 
two linearly independent solutions for the first DE.  there's a lot of 
algebra to be done - the computer's very handy for this.  note that the 
directions don't explicitly ask you to use a computer; if you really want 
to do both problems by hand, you're welcome to...

> However, rsolve is very difficult to use

"rsolve" is used to find an explicit solution to a set of recurrence 
relations.  you won't need it here.

> Help on syntax would be appreciated as well, this 
> problem is extremely hard to follow.  

well, i'm guessing you mean the solution procedure is difficult to follow.

an overview:

first, we want to find the behavior of the solution near the singular 
point (because that's where we're centering our series, and series get all 
their information from the behavior near the center).

to do this, we find the solution to a related differential equation (an 
Euler-ized version of the original) that has the same behavior near the 
singularity.

once we have the solution to Euler-ized problem, in this case:
  y_Euler = C1 x^0  +  C2 x^(1/2),
we form the solution to the original problem by substituting series in 
place for the constants above:
  y_Original = ( \sum_0^{\Infinity} a_n x^n ) x^0  +
               ( \sum_0^{\Infinity} b_n x^n ) x^(1/2)
note there're two different series with two different sets of coefficients 
there.

for solution corresponding to the larger root (1/2 in this case), the 
recurrence relation has already been worked out (formula #8 on p274 - 
replace "r" with "1/2").

since the roots of the indicial equation - 0 and 1/2 - don't differ by an 
integer, this same recurrence relation will also work for the smaller 
root.

since you're asked for six terms, applying this formula by hand can be a 
pain.  here's a good use for maple...

if you have questions about syntax, let me know.  please be specific,
though ("i tried this, and got that message" or "i can't figure out a
function to do this operation").

hope this helps.  lemme know if you have more questions.

j


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