  We study a class of linear ordinary differential equations (ODE)s with
distributional coefficients. These equations are defined using an {\it
intrinsic} multiplicative product of Schwartz distributions which is an
extension of the H\"ormander product of distributions with non-intersecting
singular supports [L. H\"ormander, The Analysis of Linear Partial
Diffe\-rential Operators I, Springer-Verlag, 1983]. We provide a regularization
procedure for these ODEs and prove an existence and uniqueness theorem for
their solutions. We also determine the conditions for which the solutions are
regular and distributional. These results are used to study the Euler-Bernoulli
beam equation with discontinuous and singular coefficients. This problem was
addressed in the past using intrinsic products (under some restrictive
conditions) and the Colombeau formalism (in the general case). Here we present
a new intrinsic formulation that is simpler and more general. As an
application, the case of a non-uniform static beam displaying structural cracks
is discussed in some detail.
