A perturbation scheme is devised for investigating the periodic solutions of a nonlinear, second‐order, ordinary differential equation with an additional small nonlinear term. It is assumed that when the additional term vanishes the solution to the remaining zeroth‐order nonlinear equation is known. When this is the case, explicit expressions for the first‐order solution and the first‐order correction to the dispersion relation are given in terms of the known zeroth‐order solution.