Linear ordinary differential equations L(y) = b (where L(y) :9Q y(n) + an − 1:- Cy(n − 1) + . . . a0y) are studied. It is proved, in Theorem 1, that a result valid for n = 2 can be extended for an arbitrary n. After that, Theorem 2, related in some sense to a classical Liouville theorem, is formulated and proved. Furthermore, Theorem 2 is used to achieve an elementary version (Theorem 3) of the first result (Theorem 1). Finally, several open problems are listed.
The theorems proved in this paper are not constructive (or, at least, it is not shown how these theorems can be applied in actual computations connected with the solution of ordinary differential equations). By these theorems, the authors have established existence of solutions with certain properties.
