Linear differential-algebraic equations perturbed by Volterra integral operators

We study linear inhomogeneous vector ordinary differential equations of arbitrary order in which the matrix multiplying the highest derivative of the unknown vector function is singular in the domain where the equations are defined. We also study perturbations (not necessarily small) of such equations, which are linear integro-differential equations with a Volterra operator. We obtain sufficient conditions for the solvability of such equations and give representations of their general solutions; solvability and uniqueness conditions are also given for initial value problems for such equations. The influence of small perturbations of the free term and the initial data on the solution is considered. A numerical method is suggested. The results of numerical experiments are given.