On Expansion of a Solution of General Non-autonomous Polynomial Differential Equation

We give a recursive formula for an expansion of a solution of a general non-autonomous polynomial differential equation. The formula is given on the algebraic level with a use of shuffle product. This approach minimizes the number of integrations on each order of expansion. Using combinatorics of trees we estimate the radius of convergence of the expansion.Comment: 13 pages, To be published in Journal of Dynamical and Control System