Classical, analytical, formal and generalized solutions of a first-order differential equation with a meromorphic coefficient

The paper considers the question of the existence of generalized solutions of a homogeneous linear differential equation of the first order with a generalized coefficient. The case is investigated when the generalized coefficient coincides with a given meromorphic function on the complement to the set of poles of this function and, moreover, the corresponding equation on the complex plane has a meromorphic solution. All generalized functions are described that coincide with the considered meromorphic function on the complement to the set of poles. For equations with such generalized coefficients, the concept of a formal solution is introduced, and such solutions are constructed in an explicit form.The main result consists in describing those generalized coefficients from the class under consideration, for which there is a generalized solution.