A refinement of Kovalevskaya's theorem on analytic solvability of the Cauchy problem

In this paper we give a proof of an analog of the Kovalevskaya theorem about analytic solvability of the Cauchy problem for a linear differential equation with constant coefficients. A major role in the proof is played by the Borel transform and the Laurent expansion of the function $P^{-1}$, where $P$ is the characteristic polynomial. This expansion produces an efficiently computable approximation of the solution of the Cauchy problem. The method of the proof allows to consider equations not necessarily resolved with respect to the highest derivative, however it imposes additional restrictions on the right hand side.