Differential Transcendency in the Theory of Linear Differential Systems with Constant Coefficients

Abstract

In this paper we consider a reduction of a non-homogeneous linear system of first order operator equations to a totally reduced system. Obtained results are applied to Cauchy problem for linear differential systems with constant coefficients and to the question of differential transcendency. Keywords:Linear system of first order operator equations with constant coefficients, n th order linear operator equation with constant coefficients, the characteristic polynomial, sum of principal minors, differential transcendency Email addresses : Branko Malešević <malesevic@etf.rs> , Dragana Todorić <draganat@math.rs> , Ivana Jovović <ivana@etf.rs> , Sonja Telebaković <sonjat@math.rs>

A note on the reduction formulas for some systems of linear operator equations

“…At the beginning of this section we first present two standard lemmas for reduction process using canonical forms. Afterwards we give a brief exposition of two main results from [5] and [6]. where ν = T −1 ϕ is its nonhomogeneous term and z = T −1 x is a column of the unknowns.…”

“…Theorem 3.6. (Theorem 4.3 from [6]) Let δ i k B; A n−k ( ϕ) stands for the sum of the principal minors of the order k containing the entries of the i th column of the matrix obtained from the matrix B by replacing column A n−k ( ϕ) in the place of the i th column of B. Then the linear system of the operator equations (1) implies the system, which consists of the higher order operator equations as follows…”

“…This system consists of a higher order operator equations which differ only in the variables and in the nonhomogeneous terms. The general reduction formulas from [5] and [6] can be applied to some special systems. An example of total reduction of the linear system of the operator equations with the system matrix in the companion form can be found in [4].…”

Total reduction of linear systems of operator equations with the system matrix in the companion form

“…Each of the subsystems has the same form as the above partially reduced system, and therefore the corresponding totally reduced system is a conjunction of totally reduced systems. For the proof we refer the reader to [7].…”

“…The function x 0 is differentially transcendental over C if and only if the function ϕ is differentially transcendental over C, (see [7,10]). Since all equations of system (4.2) are of this form, our main task is to determine the conditions on which a differentially transcendental component of the nonhomogeneous term of system (4.1) does not appear in the nonhomogeneous parts of system (4.2).…”