Conditional symmetries of nonlinear third-order ordinary differential equations (Q1690358)

The starting point of this study is the following Definition 1: A nonlinear third-order ODE is conditionally classifiable by a symmetry algebra \(\mathcal{A}\) with respect to a nonlinear second-order ODE called the root ODE if the third-order ODE jointly with the second-order ODE forms an overdetermined compatible system (so the solutions of the third-order ODE reduces to the solutions of the second-order ODE) and the second-order ODE has the symmetry algebra \(\mathcal{A}\) which is called the conditional symmetry algebra of the third-order ODE. An algorithm is presented for computing the conditional symmetries of a scalar third-order ODE of the form \(y^{\prime \prime \prime }=f(x, y, y^{\prime }, y^{\prime \prime })\) subject to the nonlinear second-order ODE \(y^{\prime \prime}=g(x, y, y^{\prime })\). Several examples are classified. A main result is that the derived nonlinear ODEs without substitution have in general less inherited symmetries as compared to that of the root second-order ODEs.