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We construct a generalization of the multiplicative product of distributionspresented by L. H\"ormander in [L. H\"ormander, {\it The analysis of linearpartial differential operators I} (Springer-Verlag, 1983)]. The new product isdefined in the vector space ${\mathcal A}(\bkR)$ of piecewise smooth functions$f: \bkR \to \bkC$ and all their (distributional) derivatives. It isassociative, satisfies the Leibniz rule and reproduces the usual pointwiseproduct of functions for regular distributions in ${\mathcal A}(\bkR)$. Endowedwith this product, the space ${\mathcal A}(\bkR)$ becomes a differentialassociative algebra of generalized functions. By working in the new ${\mathcalA}(\bkR)$-setting we determine a method for transforming an ordinary lineardifferential equation with general solution $\psi$ into another, ordinarylinear differential equation, with general solution $\chi_{\Omega} \psi$, where$\chi_{\Omega}$ is the characteristic function of some prescribed interval$\Omega \subset \bkR$.

A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients