On the Explicit Solution of the Linear First Order Cauchy Problem with Distributional Coefficients

C.O.R. Sarrico

Abstract: In [3] we have considered the $n^{{\rm th}}$ order linear Cauchy problem for a class of differential equations with distributional coefficients. We have extended the concept of solution of this problem and we have proved that these solutions are consistent with the classical solutions. Here we give necessary and sufficient conditions for existence, in this extended sense, of a solution of the problem $X'=UX+V$, $X(t_{0})=a$, where $U\in C^{\infty}\oplus\Dd_{m}^{p}$ ($\Dd_{m}^{p}=\Dd^{p}\cap\Dd_{m}$, $\Dd_{m}$ is the space of distributions with nowhere dense support, $\Dd^{p}$ is the space of distributions of order $\le p$ in the Schwartz setting), $V\in\Dd$, $a\in\C$ and $t_{0}\in\R$. We also give an explicit and practical formula for computing this solution.

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