Abstract <p>An approximate method of solving the Cauchy problem for nonlinear second-order ordinary differential equations is considered. The method is based on conforming application of the shifted Chebyshev series and a Markov quadrature formula. An algorithm is briefly discussed to partition the integration interval into elementary subintervals where an approximate solution and its derivatives are represented by partial sums of shifted Chebyshev series with a prescribed accuracy. The efficiency of the proposed method is illustrated by solving nonlinear oscillations of mathematical pendulum equation. A number of advantages of the proposed method over the well-known Stoermer method of solving ordinary differential equations are analyzed.</p>