Hi,
> What would be required to support derivatives like this (and other
> related functions such as Dirac delta and Heaviside step function)?
It might be an idea to look at the work of Martin von Mohrenschildt[1,2].
He did some interesting extensions and implemented that in Maple[3]. It
might be not that difficult to implement the PPDR (piecewise polynomial
differential ring) in Axiom. I'm not sure if this exactly matches your
initial question. It would in any way be nice to make such computations
in Axiom.
Citing the abstract of the paper:
In this paper we generalize the basic notations of the Liouville-Ritt-Risch
theory of closed-form solutions to discontinuous field extensions. Our aim
is to extend the theory of differential fields such that the “classical
algorithm”
like the Risch structure theorem and the algorithm solving the Risch
differential
equation can be extended to handle discontinuous extensions.
[1]: http://e-collection.library.ethz.ch/view/eth:39463?lang=de
[2]: http://link.springer.com/chapter/10.1007%2F978-1-4612-4268-0_7
[3]:
http://www.maplesoft.com/support/helpJP/Maple/view.aspx?path=dsolve/piecewise
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