Difference scheme on an uniform mesh for a singularly perturbed Cauchy problem

Cauchy problem for a singularly perturbed second order ordinary differential equation is considered. On a base of introduced maximum principle for a Cauchy problem the solution and its derivateves are estimated. Exponential fitted scheme, generalized well-known Il'in scheme for a case of an initial value problem, is constructed. The uniform convergence of constructed scheme with the first order of an accuracy is proved. Numerical results are discussed.