Cramer's rule for implicit linear differential equations over a non-Archimedean ring

Abstract

We consider a linear nonhomogeneous $m$-th order differential equation in a ring of formal power series with coefficients from some field of characteristic zero. This equation has infinite many solutions in this ring -- one for each initial condition of the corresponding Cauchy problem. These solutions can be found using classical methods of differential equation theory. Let us suppose the coefficients of the equation and the coefficients of nonhomogeneity belong to some integral domain $K$. We are looking for a solution in the form of a formal power series with coefficients from this integral domain. The methods of classical theory do not allow us to find out whether there exists an initial condition that corresponds to the solution of the coefficients from $K$ and do not allow find this initial condition. To solve this problem, we use the method proposed by U. Broggi. This method allows to find a formal solution of the linear nonhomogeneous differential equation in the form of some special series. In previous articles, sufficient conditions for the existence and uniqueness of a solution were found for a certain class of rings $K$ with a non-Archimedean valuation. If these conditions hold, the formal power series obtained using the Broggi’s method is considered. Its coefficients are the sums of series that converge in the non-Archimedean topology considered. It is shown that this series is the solution from $K[[x]]$ of our equation. Note that this equation over a ring of formal power series can be considered as an infinite linear system of equations with respect to the coefficients of unknown formal power series. In this article it is proved that this system can be solved by some analogue of Cramer's method, in which the determinants of infinite matrices are found as limits of some finite determinants in the non-Archimedean topology.