Summary: The exact solution of a system of two second-order nonlinear ordinary differential equations is obtained in the reproducing kernel space \(W^3_2[0, 1]\). The exact solution is represented in the form of series. The \(n\)-term approximation \(u_n(x),\;v_n(x)\) are proved to converge to exact solution \(u(x),\;v(x)\). The method’s implementation requires no additional conditions, whether the equations are singular or non-singular and linear or nonlinear. Examples are presented to demonstrate the reliability and efficiency of the algorithm developed.