First order ODEs, Symmetries and Linear Transformations

TL;DR

This paper introduces an algorithm for solving first order ODEs by identifying linear symmetries, enabling the classification and solution of a broad class of equations, including most from Kamke's collection.

Contribution

The paper presents a systematic method to find linear symmetries of first order ODEs, linking them to a class of equations solvable via linear transformations.

Findings

78% of Kamke's solvable first order ODEs are in the identified class.

The algorithm can find symmetries for most equations in the class.

Partial success in identifying symmetries for Riccati equations.

Abstract

An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ xi = F(x), eta = P(x) y + Q(x) ], where xi d/dx + eta d/dy is the symmetry generator - is presented. To these {\it linear} symmetries one can associate an ODE class which embraces all first order ODEs mappable into separable through linear transformations {t = f(x), u = p(x) y + q(x)}. This single ODE class includes as members, for instance, 78% of the 552 solvable first order examples of Kamke's book. Concerning the solving of this class, a restriction on the algorithm being presented exists only in the case of Riccati type ODEs, for which linear symmetries {\it always} exist but the algorithm will succeed in finding them only partially.