The notion of Lagrangian, partial Lagrangian, Hamiltonians and partial Hamiltonians are used to obtain closed-form solutions and reduction of ordinary differential equations(ODEs) via first integrals. The partial Lagrangian and partial Hamiltonian approaches are algorithmic and systematic to compute the first integrals which can be used to derive the closed form solutions for differential equations arising in different areas of sciences, engineering, economics, finance, business etc. We utilize these elegant approaches to numerous physical models to obtain closed form solutions which provide novel ideas to develop a deep insight and room for analysis of models under considerations. The canonical Hamiltonian systems arise in almost all fields except the economic growth theory and other economic models where the current value Hamiltonian system exist. For a state constrained optimal control problem, Pontryagin’s maximum principle provide necessary optimality conditions. Due to state constraints, the resulting system of equations arise from necessary conditions are difficult to solve. We investigate effective management of HIV infection and SIER models. In the proposed research our focus will be to classify the arbitrary functions involved in optimization models as well as in system of ordinary differential equations. In order to cover gap in the literature, we shall develop new techniques to find first integrals of dynamical systems which do not admit standard Lagrangians and Hamiltonians. Then the closed form solutions will be constructed via first integrals. In addition to the above, we shall find Lie and non-classical symmetries of non-linear system of coupled differential equations which involve several parameters. The linearization of differential equation will be discussed and we shall show that how one can linearize the non-linear dynamical systems to linear models.