We discuss the closed-form solutions of an economic growth model for tourism which aims at optimizing the quality of life of residents. The current value Hamiltonian function is used to determine the necessary conditions for the optimal values of consumption and number of tourists in the destination. We show that how one can use the necessary conditions, along with the partial Hamiltonian approach to compute first integrals of the model under consideration. Then these first integrals were utilized to compute the closed-form solutions of the system. This study points out new ways of solving systems of differential equations as well as to optimize the functional along with constraints.