Ordinary boundary value problems involving an eigenvalue occur very frequently in physics, atomic physics in particular. This work is a study of numerical solutions of these problems using initial value problems techniques. A study of a numerical procedure that deals efficiently with the inherent instability of fJic problem is presented in details. Also, methods devised to a single eigenvalue problem are generalized and applied to multidimensional problems. Moreover the problems considered here assume some known asyrnptotic forms have analytical solutions for large values of the independent variable, and these forms are adopted to improve the efficiency of the algorithm. Numerical examples are included to illustrate the computational aspects of the thesis.