The aim of this book is to use a semi-analytic technique for solving singular initial value problems of ordinary differential equations with a singularity of different kinds to construct polynomial solution using two point osculatory interpolation. The efficiency and accuracy of suggested method is assessed by comparisons with exact and other approximate solutions for a wide classes of non–homogeneous,non–linear singular initial value problems. Many examples are presented to demonstrate the applicability and efficiency of the suggested method on one hand and to confirm the convergence order on the other hand,two applications in mathematical physics and astrophysics are presented,such as Lane–Emden equations and Emden–Fowler equations to model several problems such as the theory of stellar structure,the thermal behavior of a spherical cloud of gas, isothermal gas spheres and the theory of thermionic currents.