Integration of ordinary differential equations via nonlocal symmetries
It is a classical result that \(n\)th-order ordinary differential equations with a solvable Lie algebra of point symmetries \(\mathfrak{g}\) (also called fundamental algebra of the equation) can be integrated by quadratures. The structure of the algebra \(\mathfrak{g}\) can also be used to obtain other simplifications or reductions of an equation. Dealing with subalgebras which are not ideals leads to the so-called nonlocal symmetries of the reduced equation [\textit{L. S. Krasil'shchik} and \textit{A. M. Vinogradov}, Acta Appl. Math. 15, 161--209 (1989; Zbl 0692.35003)]. Nonlocal symmetries constitute an interesting approach to the analysis of differential equations, and have been succesfully applied to reduce third-order equations with a three-dimensional simple fundamental algebra to a Riccati equation [\textit{N. H. Ibragimov} and \textit{M. C. Nucci}, Lie Groups Appl. 1, 49--64 (1994; Zbl 0921.34015)]. In [\textit{A. A. Adam} and \textit{F. M. Mahomed}, IMA J. Appl. Math. 60, 187--198 (1998; Zbl 0908.34008)], the authors introduced a nonlocal symmetry method for first-order differential equations. In the paper under review, this method is adapted and applied to study second-order ordinary differential equations, provided that the equation admits the two-dimensional affine Lie algebra \(\mathfrak{r}_{2}\) (for the two types of realisations as Lie algebras of vector fields on the plane) as an algebra of point symmetries. Examples on double reduction and Abel equations [\textit{L. M. Bercovich}, Lie Groups Appl. 1, 27--37 (1994; Zbl 0926.35056)] are given. Further, an integration procedure for first-order equations admitting an exponential nonlocal symmetry under an additional constraint is given.
scientific article; zbMATH DE number 1906542