In this work it is studied the behaviour of a class of autonomous, one degree of freedom, non-linear oscillators with random coefficients and initial conditions. Analytical approximated results regarding the joint density function and the moments of the state variables are deduced by a suitable application of the classical perturbation methods used in the deterministic cases, and by making use of some constructive theorems proved in ref.s[1] and[2]. The analytical results are utilized in two simple examples, to study the behaviour of an undamped non-linear oscillator and a Van der Pol oscillator with random constant coefficients and initial conditions.