On a method to transform algebraic differential equations into universal equations (Q1766163)

By a theorem of \textit{L. A. Rubel} [AiAA J. 19, 863--871 (1981; Zbl 0458.76013)], there exists a fourth-order homogeneous algebraic ordinary differential equation, which is \(C^\infty(\mathbb{R})\)-universal, that means that any \(C^\infty\)- function on \(\mathbb{R}\) may be uniformly approximated by solutions of this differential equation. The author gives an algorithmic method to transform homogeneous algebraic differential equations into \(C^\infty\)-universal differential equations (this notion is defined in an analogous way). The procedure given by the main result (Theorem 1.1) is clear and effective and gives also as special cases some \(C^\infty\)-universal equations constructed by \textit{R. J. Duffin} [Proc. Natl. Acad. Sci. USA 78, 4661--4662 (1981; Zbl 0463.41012)]. Elimination of the parameter \(n\) by differentiation leads also to \(C^\infty\)-universal equations.