  The purpose of this paper is twofold. An immediate practical use of the
presented algorithm is its applicability to the parametric solution of
underdetermined linear ordinary differential equations (ODEs) with coefficients
that are arbitrary analytic functions in the independent variable. A second
conceptual aim is to present an algorithm that is in some sense dual to the
fundamental Euclids algorithm, and thus an alternative to the special case of a
Groebner basis algorithm as it is used for solving linear ODE-systems. In the
paper Euclids algorithm and the new `dual version' are compared and their
complementary strengths are analysed on the task of solving underdetermined
ODEs. An implementation of the described algorithm is interactively accessible
under http://lie.math.brocku.ca/crack/demo.
