Hidden symmetries associated with the projective group of nonlinear first-order ordinary differential equations

Abstract

Hidden symmetries, those not found by the classical Lie group method for point symmetries, are reported for nonlinear first-order ordinary differential equations (ODEs) which arise frequently in physical problems. These are for the special class of the eight nonAbelian, two-parameter subgroups of the eight-parameter projective group. The first-order ODEs can be transformed by non-local transformations to new separable first-order ODEs which then can be reduced to quadratures. The first-order ODEs include Riccati equations and equations which in particular cases are of the form of Abel's equation. The procedure demonstrates the feasibility of integrating nonlinear ODEs that do not show any apparent Lie group point symmetry. Applications to the Vlasov characteristic equation and the reaction-diffusion equation are given.