Classification of linear differential equations with rational function coefficients

Abstract

The basis of the classification of linear differential equations presented here is the order of growth of its solutions in a neighbourhood of the singularities. This order of growth is bounded from below by the corresponding slopes of the associated Newton Puiseux diagram. All “important” linear differential equations of second order of mathematical physics have solutions whose order of growth are equal to the NP slopes. In general, we call such linear differential equations of order n with rational function coefficients “important” which have weakly growing solutions like above. The classification presented here with the support of Newton Puiseux diagrams becomes for all “important” differential equations a classification by the growth behaviour of their solutions in a neighbourhood of their singularities and, finally, a classification by the essential characteristic of these differential equations.