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\centerline{\bf On the New Face of Kovacic's Algorithm}

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\centerline{Manuel Bronstein}
\centerline{ETH Zentrum}
\centerline{Wissenschaftliches Rechnen, IFW D.25.1}
\centerline{CH - 8092 Z\"{u}rich, Switzerland}
\centerline{Email: bronstein@inf.ethz.ch}

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Recent progress on Kovacic's algorithm has radically changed the way
in which equations of the form     y'' + p(x) y' + q(x) y = 0    are
solved symbolically. In this talk we describe the rational algorithm
of Ulmer and Weil for completely reducible second order equations, then
discuss its efficient stand-alone implementation in the bernina system, and
finally outline new rational heuristics for the non-completely reducible case.

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