In this paper we compare the inapplicable cases of the Method of Undetermined Coefficients and Method of Variation of Parameters in solving second order differential equations with constant coefficients. We first analyse the cases of in applicability for each method to identify the in applicability, investigate their respective limitations and the underlying reasons responsible for it .The applicability of Method of undetermined coefficients clearly depends upon the forcing functions on the other hand the method of variation of parameter limits itself over computational complexities over non elementary integrals. Then finally we carried out comparative analysis to highlight the differences and sameness between the limitations of these methods.