In this paper, we seek analytical approximate solutions of a problem coupling the neutronics equation and the thermal equation (using incomplete Elliptic integrals), and we compare these analytical approximate solutions to a numerical method for the coupled problem, using a Crank-Nicolson scheme for the thermal equation and an generalized eigenvalue problem for the neutronics equation. Both the approximate analytical methods and the numerical method amount to finding the multiplication factor k ef f (k −1 ef f is the generalized eigenvalue) as the unique solution of an equation. All methods yield values of k ef f close to each other with a difference of magnitude 10 −7 , which is much better than what is generally needed in neutronics.