Global solutions of first order linear systems of ordinary differential equations with distributional coefficients

With the help of our distributional product we define four types of new solutions for first order linear systems of ordinary differential equations with distributional coefficients. These solutions are defined within a convenient space of distributions and they are consistent with the classical ones. For example, it is shown that, in a certain sense, all the solutions of X₁′=(1+δ)X₁−X₂, X₂′=(2+δ′)X₁+4X₂+δ″ have the form X₁(t)=c₁(e^2t−2e^3t)−14e^3t−δ(t), X₂(t)=c₁(4e^3t−e^2t−δ(t))+28e^3t−18δ(t)+δ′(t), where c₁ is an arbitrary constant and δ is the Dirac measure concentrated at zero. In the spirit of our preceding papers (which concern ordinary and partial differential equations) and under certain conditions we also prove existence and uniqueness results for the Cauchy problem.