Rational General Solutions of High Order Non-autonomous ODEs

In this paper, we generalize the results of Ngo and Winkler [18, 20, 21] to the case of high order non-autonomous algebraic ODE with a birational parametrization of the corre- sponding algebraic hypersurface. First, we reduce the problem for finding rational general solutions of non-autonomous n -1 (n > 2) order ODE to finding rational general solutions of an associated first order rational system of autonomous ODEs in n indeterminates based on the parametrization of hypersurface. Next, the correspondence of the rational general solutions between the original non-autonomous algebraic ODE and the associated system of autonomous ODEs is proved. Finally, a criterion is presented for existence of rational general solutions of the associated system of autonomous ODEs if the degree bound of its rational general solutions is given. Moreover, we give some nice properties of polynomial system of autonomous ODEs.