Summary: The problems of construction and theoretical substantiation of numerical-analytical algorithms for polynomial approximation of the Cauchy problem solutions for Abel’s differential equation are considered. The proposed algorithm is based on the V. K. Dzyadyk, Approksimatsionnye metody resheniya differentsial’nykh i integral’nykh uravnenij. Kiev: Naukova Dumka (1988; Zbl 0708.65067) approximation method for the solution of differential and integral equations, the main idea of which is to construct such an approximate solution that would satisfy the Chebyshov approximation theorem on the characterization of the best approximation polynomial as accurately as possible. In the paper the \(a\)-method is generalized to equations with nonlinearities in the form of polynomials. A theorem on the deviation of the approximate solution from the exact solution of the given Cauchy problem for uniform and quadratic metrics is proved, the estimations of errors are obtained. The algorithm was tested on a test task. The computational experiment illustrates the high efficiency of the proposed algorithm and theoretical results.