Linearly Satellite Unknowns in Linear Differential Systems

Abstract

Let K be a differential field of characteristic 0. Consider a linear differential system S of the form $y'=Ay$, where $A\in K^{n\times n}$ and $y=(y_1,\ldots ,y_n)^T$ is a vector of unknowns. In the present work we introduce a concept of linearly satellite unknowns: for the nonempty set of selected unknowns $s=\{y_{i_1},\ldots ,y_{i_k}\}$ an unselected unknown $y_j$ is called linearly satellite if the j-th component of any solution to S can be linearly expressed over K only via selected components of this solution and their derivatives. We present an algorithm for linearly satellite unknown recognition and its implementation in Maple. The ability to determine linearly satellite unknowns can be used for partial solving of differential systems.