Is there a numerical ODE solver facility in Maxima / off track -- symbolic ODE solver for series

Richard Fateman wrote:
> Sheldon Newhouse wrote: 
>   
>> ......
>>     
>
>
>   
>> Since your ODE has the form y'' = h(x,y) where h is analytic in a 
>> neighborhood of (x,y) = (0,0) (actually a polynomial), the solution 
>> y=y(x) is analytic in a neighborhood of x=0. So there is a convergent 
>> power series representation (its radius of convergence is positive, 
>> but I don't know how big (it may be known how to estimate this, but I 
>> don't know it. Normally, the majorant method is used for this, and it 
>> is dependent on the structure of h(x,y). Note that even if h(x,y) is a 
>> polynomial, then the solution may be not be an entire function:
>> e.g y' = y^2, y(0)=1 has solution y = -1/(x-1) ) 
>>     
> .....  I think the analysis that this illustrates is approximately the 
> kind of thing that should be done by an  ODE solver in a CAS.
> That is, a qualitative description of what is going on, other info, and 
> {if appropriate, convenient, ...} a solution.
>
> This is not something I ordinarily consume (solutions of ODEs) in any 
> part of my life, at least since a time when I taught sophomore ODE
> course.  But my understanding is that people would really like to know 
> the characteristics of the solution(s). Even how many of them or
> if there are degenerate ones;  in a parameterized equation, where the 
> solutions exist, singularities, and who knows what.  Maybe even
> suggestions about numerical solution methods.  Like this Taylor-series 
> style should not work for stiff differential equations (how stiff?).
>
> Or maybe just "this ODE is of type XYZ discussed in this monograph, and 
> can be transformed into the canonical form Q this way..."
>
>
> Anyway, consider instead of "ODEsolve"  how about "ODEdescribe"?
>
> As I said, this is not something I use myself, so perhaps others should 
> fill out the recipe ingredients for ODEdescribe. SEN perhaps?
>
> RJF
>
>
>
>
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>   
Actually, this is quite an interesting topic. I think that CAS has a lot 
of potential for the study of differential equations. The current 'ode2' 
program is sufficient for much, (but not all) of the first order 
equations in a standard sophomore level course in ODE's. It seems that 
Maxima is a long way from Maple in this respect. Chapter 17 in the book 
"Introduction to Maple, 3rd edition by Andre Heck (pp 521-618) has a lot 
of interesting material --including references. There also is a nice 
book by Gray-Mezzino-Pinsky on the use of Mathematica for a sophomore 
level ODE class. It would be nice to at least port the tools in those 
books to Maxima. I'll look into this and I encourage any other 
interested people to do the same.

-sen