Implementation of the Runge - Kutta method for solving odes with a discontinuous right-hand side

In this paper considers the solution of the Cauchy problem for first-order ODEs with a second-order discontinuity by the variable x on the right-hand side using the Runge - Kutta method. The standard Karateodori and Filippov approaches are not applicable for numerical calculations of such problems. An analysis of the solutions the numerical examples showes that the Runge - Kutta method should be applied with decreasing the mesh in the neighborhood of the discontinuity point according to a certain rule proposed in the paper.