Factorization method for solving systems of second-order linear ordinary differential equations - Parasidis, I.N.; Providas, E. (Eurasian Mathematical Journal, 2025)
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), Φ(X) = _j=1^n M_j Ψ_j(X) and X’’(t) - S(t)X’(t) - Q(t)X(t) = F(t), Φ(X) = _j=1^n M_j Ψ_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.