On solutions of polynomial growth of ordinary differential equations
In this extensive paper the author presents a theorem (main theorem) on the existence of solutions of polynomial growth of ordinary differential equations of type \(E:dY/dX=F(X,Y)\), where F is of class \(C^ 1\). The asymptotic behaviour of these solutions and the variation of neighbouring solutions are obtained by solving an asymptotic functional equation related to E. The proof of the main theorem uses methods of nonstandard analysis. By a linear, nonstandard change of variables, called a macroscope, the original equation is transformed into a slow-fast differential equation, or singular perturbation. The existence is then proved using nonstandard perturbation theory. The result is an extension of results by F. and M. Diener and G. Reeb on solutions of polynomial growth of rational differential equations.
scientific article; zbMATH DE number 4146471
scientific article