A general solution of a linear dissipative oscillatory system avoiding decomposition into eigenvectors

An algorithm for constructing a general solution of an oscillatory system without determining the eigenvectors, which are the main source of the computational instability of classical methods, is proposed and justified. Souriau's algorithm [1-3], developed for solving systems of linear algebraic equations, is extended to the construction of a general solution of systems of linear ordinary differential equations. The adjoint matrix, consisting of the cofactors of the elements of the initial matrix, and the representation of the characteristic polynomial in terms of the characteristic adjoint matrix, play an important role here. the algorithm consists of simple algebraic operations, in addition to the numerical integration of a single differential equation, called the characteristic equation.