Abstract:
The Cauchy problem for a first-order differential equations linear inhomogeneous system of with a random Gaussian perturbation and random inhomogeneity considered. Constructed an auxiliary deterministic differential equations linear system , containing ordinary and variational derivatives, with a deterministic initial condition. The deterministic Cauchy problem solution allows one to obtain a formula for the solution mathematical expectation to the original Cauchy problem. We found periodic on the mean system solutions existence conditions and an explicit formula for the periodic mathematical expectation. A similar technique is used to obtain a deterministic problem that allows finding an explicit formula for the second moment function solution to the original Cauchy problem. Second moment function solution periodicity conditions and an explicit formula for the periodic second moment function are also obtained.
Keywords:linear differential equations systems, variational derivative, mathematical expectation, periodic on mean solution, second moment function.
