  We develop a $D-$module approach to various kinds of solutions to several
classes of important differential equations by long divisions of different
differential operators. The zeros of remainder maps of such long divisions are
handled by an analogue of Hensel's lemma established recently from valuation
theory. In particular, this explains the common origin of some classically
known special function series solutions of Heun equations and usual Frobenius
series solutions. Moreover, these remainder maps also generate eigenvalue
problems that lead to non-trivial factorizations of certain generalized
hypergeometric operators.
