Numerical integration of the Cauchy problem with non-singular special points

Solutions of many applied Cauchy problems for ordinary differential equations have one or more multiple zeros on the integration segment. Examples are special functions mathematical physics. The presence multiples significantly complicates numerical calculation, since such ill-conditioned. Round-off errors may corrupt all decimal digits solution. Therefore, should be treated as points equations. In present paper, a local solution transformation is proposed, which converts zero into simple one. calculation latter not difficult. This makes it possible to dramatically improve accuracy and reliability calculation. Illustrative examples been carried out, confirm advantages proposed method.