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

Richard Fateman fateman at cs.berkeley.edu
Sun Apr 5 09:08:57 CDT 2009

```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..."