The transcendental Risch differential equation
Let \(k\) be a differential field of characteristic \(0\) and \(K\) be a differential field extension of \(k\) (with derivation \(x\mapsto x': K\to K\)); let \(\theta\) be an element of \(K\) satisfying the following conditions: \(\theta\) is transcendental over \(k\), \(k\) and \(k(\theta)\) have the same field of constants, and either \(\theta\) is primitive over \(k\) (i.e. \(\theta '\in k)\) or \(\theta\) is exponential over \(k\) (i.e. \(\theta '/\theta =\eta '\) for some \(\eta\in k\)). A differential equation \[ y'+fy=g\text{ with } f,g\in k(\theta) \tag{*} \] is called a Risch differential equation. There is a considerable interest in solving such a differential equation, because algorithms for the integration of elementary functions in finite terms rely on the resolution of Risch differential equations. In 1969 \textit{R. H. Risch} published an algorithm for solving (*) [cf. Trans. Am. Math. Soc. 139, 167--189 (1969; Zbl 0184.06702)] which required factoring the denominators of \(f\) and \(g\). Later, \textit{M. Rothstein} [Ph. D. Thesis, Univ. of Wisconsin, Madison, 1976] and \textit{J. H. Davenport} [SIAM J. Comput. 15, 903--918 (1986; Zbl 0632.65091)] published algorithms for solving (*) which use square-free decompositions of the denominators of \(f\) and \(g\). Davenport's algorithm has been implemented in the computer algebra systems Scratchpad II and Maple; however it requires \(f\) to be in a certain form (called weakly normalized), but no complete algorithm that makes \(f\) weakly normalized has been published. The author describes a new algorithm for solving equation (*) or for showing that it doesn't have a solution. This algorithm requires \(f\) to have an integral which is elementary over \(k(\theta)\) and reduces the problem to the same problem for an equation \(AQ'+BQ=C\) with elements \(A,B,C\) of \(k[\theta]\) if \(\theta\) is primitive over \(k\), resp. with elements \(A\in k[\theta]\) and \(B,C\in k[\theta,\theta^{-1}]\) if \(\theta\) is exponential over \(k\). Then an algorithm of Rothstein is used to decide whether there exists a solution of this equation and to obtain one in case of existence. In the last two sections of the paper the author describes Rothstein's algorithm in detail and makes some remarks on an implementation in Scratchpad II.
scientific article; zbMATH DE number 4160893