Factorization method for solving systems of second-order linear ordinary differential equations

Abstract

We consider in a Banach space the following two abstract systems of first-order and second-order linear ordinary differential equations with general boundary conditions, respectively,

X'(t) - A_0(t) X(t) = F(t),  \Phi(X) = \sum_{j=1}^{n} M_j \,\Psi_j(X)

and

X''(t) - S(t)X'(t) - Q(t)X(t) = F(t),    \Phi(X) = \sum_{j=1}^{n} M_j \Psi_j(X), 

where X(t)=col(x_1 (t),…,x_m (t)) denotes a vector of unknown functions, F(t) is a given vector and A_0 (t),  S(t),  Q(t)  are given matrices, Φ, Ψ_1, . . . , Ψ_n, Θ_1, . . . , Θ_r   are vectors of linear bounded functionals, and M_1, . . . , M_n, C, N_1, . . . , N_r are constant matrices. We first provide solvability conditions and a solution formula for the first-order system. Then we construct in closed form the solution of a special system of  2m first-order linear ordinary differential equations with constant coefficients when the solution of the associated system of  m first-order linear ordinary differential equations is known. Finally, we construct in closed form the solution of the second-order system in the case in which it can be factorized into first-order systems.