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We study a class of linear ordinary differential equations (ODE)s withdistributional coefficients. These equations are defined using an {\itintrinsic} multiplicative product of Schwartz distributions which is anextension of the H\"ormander product of distributions with non-intersectingsingular supports [L. H\"ormander, The Analysis of Linear PartialDiffe\-rential Operators I, Springer-Verlag, 1983]. We provide a regularizationprocedure for these ODEs and prove an existence and uniqueness theorem fortheir solutions. We also determine the conditions for which the solutions areregular and distributional. These results are used to study the Euler-Bernoullibeam equation with discontinuous and singular coefficients. This problem wasaddressed in the past using intrinsic products (under some restrictiveconditions) and the Colombeau formalism (in the general case). Here we presenta new intrinsic formulation that is simpler and more general. As anapplication, the case of a non-uniform static beam displaying structural cracksis discussed in some detail.

Ordinary differential equations with singular coefficients: an intrinsic formulation with applications to the Euler-Bernoulli beam equation