Application of the Smoothing Method to a Stochastic Ordinary Differential Equation

We study the use of the ``smoothing method'' to calculate the second-order moments of solutions of stochastic, ordinary, linear differential equations. We consider in detail the equation d2udz2+β02[1+ηN(z)]u=0,where N(z) is a real, zero mean, wide-sense stationary stochastic process and β0 and η ≪ 1 are positive constants. We show that, for one choice of N(z), the conventional use of the smoothing method yields correct first-order moments of the solutions, but badly incorrect second-order moments. We develop what we believe is a better way to use the smoothing method to calculate second-order moments. For the special choice of N(z), this method yields exact results. The method can be extended to the calculation of moments of all orders for arbitrary stochastic, ordinary, linear differential equations.