The weak damped oscillation in a random medium (Q1066559)

The paper deals with the random differential equation \[ \ddot x(t)+K^ 2(t)x(t)=-\epsilon f(\dot x(t),t) \] where f is a function satisfying certain conditions and K is of the form \(K(t)=K_ 0[1+\epsilon V_ 1(t)+\epsilon^ 2V_ 2(t)+...]\) with \(K_ 0=const\) and \(V_ j(t)\), \(j=1,2,..\). are random processes. The authors seek a solution in the form \[ x(t)=x_ 0(t)+\epsilon x_ 1(t)+\epsilon^ 2x_ 2(t)+... \] and derive recursive formulas for the \(x_ j(t)\) and for the mean and the correlation function.