The author considers the problem of numerical solving the equation
with and the initial condition . The origin integro-differential equation is chained on two integral equations with respect to functions and . The numerical method for solving the obtained system is a Legendre-collocation method with Gauss quadrature formulas for integral terems proposed by T. Tang, X. Xu and J. Cheng [J. Comput. Math. 26, No. 6, 825–837 (2008; Zbl 1174.65058)]. A theoretical -estimation of error of the required solution (via it’s different norms) is derived. A numerical example illustrates the theoretical results.