First-order ordinary differential equations, symmetries and linear transformations

Abstract

We present an algorithm for solving first-order ordinary differential equations by systematically determining symmetries of the form $[\xi=F(x),\, \eta=P(x)\,y+Q(x)]$, where $\xi\; \pa/\pa x + \eta\; \pa/\pa y$ is the symmetry generator. To these linear symmetries one can associate an ordinary differential equation class which embraces all first-order equations mappable into separable ones through linear transformations $\{t=f(x),\,u=p(x)\,y+q(x)\}$. This single class includes as members, for instance, 429 of the 552 solvable first-order examples of Kamke's [12] book. Concerning the solution of this class, a restriction on the algorithm being presented exists, only in the case of Riccati equations, for which linear symmetries always exist, but the algorithm will only partially succeed in finding them.