Response and Stability of a Random Differential Equation: Part I—Moment Equation Method

A linear stochastic differential equation of order N excited by an external random force and whose coefficients are white noise random processes is studied. The external force may be either white or colored noise random process. Given the statistical properties of the coefficients and of the force, equivalent statistics are obtained for the response. The present solution method is based on the derivation of the equation governing the response autocorrelation function. The simplifying assumption that the response is stationary when the coefficients and input force are stationary is introduced. Another simplification occurs with the assumption that the response is uncorrelated from the random coefficients. Closed-form solutions for the response autocorrelation function and spectral density are derived in conjunction with a stability bound.