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Rational Heuristics for Rational Solutions of Riccati Equations

by Manuel Bronstein, Thom Mulders
"... We describe some new algorithm and heuristics for computing the polynomial and rational solutions of bounded degree of a class of ordinary differential equations, which includes generalized Riccati equations. As a consequence, our methods can be used for factoring linear ordinary differential equati ..."
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equations. Since they generate systems of algebraic equations in at most n unknowns, where n is the order of the differential equation, they are particulary effective for first-order Riccati equations, which is confirmed by experimental timings for that case. Combined with the Ulmer-Weil algorithm and a

Liouvillian solutions of linear differential equations of order three and higher

by Mark Van Hoeij, Jean-francois Weil, Felix Ulmer, Jacques-arthur Weil - JOURNAL OF SYMBOLIC COMPUTATION , 1999
"... Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closed-form”) solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van ..."
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coefficient. These coefficients come “for free ” as a byproduct of our algorithm for computing semi-invariants. We specifically detail the algorithm in the cases of equations of order three (order two equations are handled by the algorithm of Kovacic (1986), see also Ulmer and Weil (1996) or Fakler (1997
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