A complete set of first integrals for any third-order ordinary differential equation admitting a Lie symmetry algebra isomorphic to is explicitly computed. These first integrals are derived from two linearly independent solutions of a linear second-order ODE, without additional integration. The general solution in parametric form can be obtained by using the computed first integrals. The study includes a parallel analysis of the four inequivalent realizations of , and it is applied to several particular examples. These include the generalized Chazy equation, as well as an example of an equation which admits the most complicated of the four inequivalent realizations.
A novel procedure to reduce by four the order of Euler-Lagrange equations associated to n-th order variational problems involving single variable integrals is presented. In preparation, a new formula for the commutator of two C ∞symmetries is established. The method is based on a pair of variational C ∞-symmetries whose commutators satisfy a certain solvability condition. It allows one to recover a (2n − 2)-parameter family of solutions for the original 2n-th order Euler-Lagrange equation by solving two successive first order ordinary differential equations from the solution of the reduced Euler-Lagrange equation. The procedure is illustrated by two different examples.
Abstract. Third-order ordinary differential equations with Lie symmetry algebras isomorphic to the nonsolvable algebra sl(2, R) admit solvable structures. These solvable structures can be constructed by using the basis elements of these algebras. Once the solvable structures are known, the given equation can be integrated by quadratures as in the case of solvable symmetry algebras.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.