This contribution applies the stochastic equivalent linearization technique to derive accurate approximations of key statistical measures -such as mean, variance, correlation, and higher moments—for a specific class of perturbed linear oscillators. These oscillators are characterized by a nonlinear term that depends on both position and velocity (cross-nonlinearity) and are driven by a zero-mean stationary Gaussian stochastic process. Utilizing this essential probabilistic information, we then apply the principle of maximum entropy to develop non-Gaussian approximations of the solution’s probability density function. Finally, we present several numerical experiments to validate the consistency of our theoretical findings.