Regarding polynomial approximation for ordinary differential equations

In this article, we consider an application of the approximate iterative method of Dzyadyk [V.K. Dzyadyk, Approximation methods for solutions of differential and integral equations, VSP, Utrecht, The Netherlands, 1995] to the construction of approximate polynomial solutions of ordinary differential equations. We illustrate that this method allows construction of polynomials of low degree with sufficiently high accuracy by examples, and as a result such polynomials can be used in practical applications. Moreover, Dzyadyk’s method produces an a priori estimate for the polynomial approximation of the solution of Cauchy problems. For the application of this method a Cauchy problem should be rewritten as the corresponding integral equation, followed by the replacement of the integrand by its Lagrange interpolation polynomial and Picard iterations.