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We prove the well-posedness of the Cauchy problem for the linear differentialsystem of the form $x^{\prime}-A(t)x=f$, where $f$ is a distribution and $A$possesses at most first-kind discontinuities together with all its derivativesdefined almost everywhere. The left-hand side of this system contains theproduct of a distribution and, in general, a discontinuous function, which isundefined in the classical space of the distributions with the smooth testfunctions $\mathcal D'$, so the Cauchy problem has no solution in $\mathcalD'$. In what follows, we cosider this system in the space of distributions withthe discontinuous test functions, whose elements admit continuous andassociative multiplication by functions possessing at most first-kinddiscontinuities (together with all their derivatives defined almosteverywhere), and show that there exists the unique solution of the Cauchyproblem which depends continuously on $f$.

On well-posedness of the linear Cauchy problem with the distributional right-hand side and discontinuous coefficients