  In this paper we generalize the involutive methods and algorithms devised for
polynomial ideals to differential ones generated by a finite set of linear
differential polynomials in the differential polynomial ring over a zero
characteristic differential field. Given a ranking of derivative terms and an
involutive division, we formulate the involutivity conditions which form a
basis of involutive algorithms. We present an algorithm for computation of a
minimal involutive differential basis. Its correctness and termination hold for
any constructive and noetherian involutive division. As two important
applications we consider posing of an initial value problem for a linear
differential system providing uniqueness of its solution and the Lie symmetry
analysis of nonlinear differential equations. In particular, this allows to
determine the structure of arbitrariness in general solution of linear systems
and thereby to find the size of symmetry group.
