First integrals of fin equations for straight fins

First integrals admitted by second-order nonlinear ordinary differential equations modeling the temperature distribution in a straight fin are obtained. After imposing the boundary conditions these first integrals give a relationship between temperature at the fin tip and the temperature gradient at the base of the fin in terms of the fin parameters. These first integrals are plotted and analyzed. The results obtained show how the temperature at the fin tip can be controlled by the temperature gradient at the base for fixed fin parameters.