Ordinary differential equation system of non-canonical shape. I

Equation $Аy^{\prime}+By=f(x)$, is considered, where $A$ and $B$ are square matrices. The solution is sought in such a class of vector functions, whose components with their derivatives in infinition increase not faster than any degree of $t$. Many authors have considered the case $\det A\neq 0$ [1–4]. In this paper the same case is considered but by means of a method, which allows to get easily controllable conditions for the solution of initial and general initial problems.