A new method to obtain either first- or second-order reductions for parametric polynomial ODEs.

For a nth-order ordinary differential equation that is either polynomial on the dependent variable and its derivatives or polynomial in the derivatives of the dependent variable, a new method to obtain either first- or second-order reductions, respectively, is introduced. The corresponding reduced equations are generalized polynomials in the dependent variable or in its derivative. The method can be applied to equations that may have integer parameters in the exponents of the polynomial of the initial equation. A procedure to determine, in function of the parameters, if the given equation may admit such reductions and to restrict the possible variations of the exponents in the reduced equations is also provided. The introduced method strictly generalizes several methods that have recently appeared in the literature. As applications of the method, this study considers first-order reductions for a generalized Emden–Fowler equation and second-order reductions for the travelling-waves equation of a fifth-order Korteweg–de Vries equation and for a Painlevé–Chazy equation. As far as we know, some of the reported reductions are new. Several Maple programs to determine if a given polynomial equation admits any of the considered reductions and, if this is the case, to determine the reduced equations are also included.