To the Study of Various Representations of Solutions of Quasi-Differential Equations in the Form of Sums of Series and Some of Their Applications

In the theory of the matrix equation $$\dot {X} = A(t)X$$ , an important role is played by the well-known representation of the solution in the form of a series. In the case of a scalar equation of the $$n$$ th order, it is also interesting to obtain a representation of the solution in the form of a scalar series the terms of which are constructed using the coefficients of the original equation. In this paper, various representations of a fundamental system of solutions of a homogeneous quasi-differential equation of the $$n$$ th order in the form of scalar series whose terms are constructed using the coefficients of the original equation are studied. For example, in the form of such series, representations of the elements of the fundamental system of solutions of the Bessel equation considered on the interval $$[\vartheta , + \infty ),$$ where $$\vartheta > 0$$ , are constructed.