Implementation of the Prelle-Singer Method for 1ODEs
L. G. S. Duarte, S. E. S. Duarte, L. A. C. P. da Mota, J. E. F. Skea
TL;DR
The paper addresses solving first-order ODEs with the Prelle-Singer (PS) method and extends it to all Liouvillian solutions (LIS). It introduces PSolver, a Maple-based implementation that exposes intermediate PS quantities like the integrating factor $R$, the ${\cal D}$ operator, and the Basis, and adds a theoretical LIS extension based on Lie symmetries. The main contribution is a practical toolkit that solves 1ODEs with elementary functions on the integrating factor and with LIS, including challenging SELFs, often outperforming standard solvers such as Maple's dsolve. The results demonstrate broader solvability and, in LIS cases, substantial time savings, making symbolic ODE solving more powerful and transparent in Maple.
Abstract
A set of MapleV R5 software routines for solving first order ordinary differential equations (1ODEs) is presented. The package implements the Prelle-Singer Method in its original form plus its extension to include elementary functions (ELFs) on the integrating factor . The package also presents a theoretical extension to deal with all 1ODEs presenting liouvillian solutions (LIS).