Linear Differential Equation with Formal Power Series Non-Homogeneity Over a Ring with a Non-Archimedean Valuation

Abstract

Consider the linear differential equation of m-th order with constant coefficients from the valuation ring K of a non-Archimedean field. We get sufficient conditions of uniqueness and existence for the solution of this equation from K [[x]]. Also the fundamental solution from 1x K[[ 1 x ]] of the equation is obtained and it is shown that the convolution of the fundamental solution and a non-homogeneity is a unique solution of the equation.