Except for rare sporadic cases, exact stationary solutions for second order nonlinear systems under Gaussian white-noise excitations are known only for certain types of systems and only when the excitations are purely external (additive). Yet, in many engineering problems, random excitations may also be parametric (multiplicative). It is shown in this paper that the method of detailed balance developed by the physicists can be applied to obtain systematically the stationary solutions for a large class of nonlinear systems under either external random excitations or parametric random excitations, or both. Examples are given for those cases where solutions have been given previously in the literature as well as other cases where solutions are new. An unexpected result is revealed in one of the new solutions, namely, under suitable combination of the parametric and external excitations of Gaussian white noises, the response of a nonlinear system can be Gaussian.

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