Numerical integration of Cauchy problems with singularity points

We propose an effective method for solving Cauchy problem for an ordinary differential equation with multiple poles of an integer order. The method provides through calculation of a pole for both single pole and chain of poles. The method uses a special algorithm for finding the multiplicity of each pole. This multiplicity is used to define the generalized reciprocal function for which the K-th order pole of the initial function is a prime zero. Calculating such a zero is not difficult, so the proposed method provides high accuracy even near the poles. After passing this zero, the calculation of the initial function resumes. Using this method on a sequence of poles permits to find a numerical solution simultaneously with a posteriori estimation of its error. The method is illustrated with test examples.