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# ODEs, PDEs and Special Functions

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ODEs and PDEs

"Computer algebra systems have evolved into powerful solving environments for studying and solving differential equations."

Some polemical questions:

 Can a Computer Algebra system compute numerical ODE solutions as fast as for instance C or FORTRAN code ?
 Can a computer really be more useful than a good book for finding exact ODE and PDE solutions ?
 Aren't these computer algebra environments more like a black-box approach to the problem ?
 Can we really study  the "differential equations" behind a problem using Computer Algebra as we would do by hand?
 Is there something fundamentally relevant regarding ODEs and PDEs that we can only do with a computer?

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Special Functions

"Special functions, their inter-relation and representations become alive within a computer"

 Conversions between mathematical functions
 Differential Polynomial Form for non-polynomial expressions

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Conclusion

"Research and education are two things highly inter-related"

 -  Constructive learning processes are mostly based on the building of logic structures by testing conjectures and analyzing the results. The proportion between success (the conjecture solves the problem) and frustration plays an important role as an emotional (+/-) accelerating factor for the whole "learning & discovery" process. -  The simultaneous analysis of a greater number of results turns apparent the underlying logic structures more rapidly, and can strengthen the intuition unexpectedly.
 - Genuine learning processes only happen when the individual who is learning participates actively.
 - Inspiration is a function of intuition, excitement and fun, transformed into results through heavy exploration. Symbolic computation can be used with these purposes, perhaps as the most important educational and research tool available at present.

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