On a method to transform algebraic differential equations into universal equations

Abstract

The paper introduces an algorithm which transforms homogeneous algebraic differential equations into universal differential equations (in the sense of L. A. Rubel) havingCn (ℝ)-solutions. By applications of the algorithm to different initial equations some new universal differential equations are found, and all the known equations due to R. J. Duffin are rediscovered with this method. Assuming weak conditions one can find Cn(ℝ)-solutionsy of the differential equation\(ny'''' y'^2 - (3n - 2)y'y''y''' + 2(n - 1)y''^3 = 0\) close to any continuous function such that 1,\(y^{(k_1 )} (q_1 ),y^{(k_2 )} (q_2 ),....,y^{(k_s )} (q_s )\) with 0 ≤k1 <k2 < .... <ks ≤n are linearly independent over the field of real algebraic numbers at the rational points q1,...,qs.