Summary: The solution of a Lyapunov matrix differential equation with constant coefficients is based on the structure of the general solution of a nonhomogeneous linear differential equation. The existence of a particular solution of this equation in a polynomial form with matrix coefficients is proven. An algorithm for calculating matrix coefficients is suggested. The validity of this method for various dynamic matrices, particularly, those having eigenvalues with zero real part, is especially important for synthesis and analysis of systems for controlling certain moving plants. An application of this method is illustrated by analysis of the influence of noises of measurement of the angular velocity on the accuracy of determination of the orientation of a space vehicle under planar rotations.