The Noether and Hamiltonian approaches have proven to be of great deal in order to compute the conservation laws and finding closed-from solutions of models under consideration. The current research plan is commenced with an idea to implement these elegant techniques to various real-world phenomenon. The closed form solutions thus obtained will provide novel ideas to develop a deep insight and room for analysis of any physical model under discussions. The approaches being employed are compatible to apply to the numerous models from engineering, mechanics, mathematical biology and several models of wide utility requiring optimal growth. With the application of partial Hamiltonian approach followed by tedious calculations results in to construct the closed form solutions. In addition, numerous important systems of first order differential equations from epidemiology will be discussed. The ongoing work concerning the computation of exact solutions shall be extended to the field of optimal growth theory where the models and functions from several related fields demanding optimal yield will be treated in order to find the optimized solutions preserving the condition of given constraints.