The initial value problem for high order linear differential systems

In this paper we derive conditions under which the system of high order differential algebraic equations R(d/dt) w = M (d/dt) f in the unknown w admits a (unique) solution satisfying initial conditions on the w´s specified by S (d/dt) w(0) = Ta with T a real matrix and a a real vector. The conditions we derive are of algebraic nature and rely on properties of modules over the polynomial ring R[ξ]. We also describe constructive algorithms which allow to check the given conditions.