In the chapter entitled Regular perturbations at the beginning of this thesis, the basic idea of the perturbation method, illustrated by a simple algebraic equation, in which the exact solution is known upfront, has been demonstrated. By observing the physical process during the Oscillator example and using regular perturbations we reached an approximate solution which was unsatisfactory due to the occurrence of secular terms. The singular behaviour which appears in all periodic motion has been corrected by using Poincaré-Lindstedt method. In order to complement the analysis of the approximate solution, terms related to convergence and uniformity have been defined in the asymptotic analysis. In the second chapter, the singular perturbation method has been dealt with. Firstly, the term dominant balancing was introduced, allowing for the explanation of the absence of the solution in the regular perturbations. Then we studied, across examples, the graph of the exact solution to the boundary problem and introduced the following terms: boundary layer, outer layer and inner and outer approximation. In the third and last chapter we conducted an analysis of the boundary layer and brought into conformity the outer and inner approximation through matching conditions. At the very end of the thesis a general procedure for solving boundary problems has been provided.