Most of the physical laws of nature can be formulated in the language of simple and independent derived equations. Examples are the Sturm-Liouville, Navier-Stokes, and Schrodinger equations in quantum mechanics. In all these equations, physical phenomena are described in the language of derivatives of space and time. The presence of derivatives in equations describes important physical quantities. These are speed, acceleration, force, friction, flow, current, and so on. Among the problems posed for differential equations, a class of logically posed problems is of particular importance, which have a solution, are the only ones that are smooth and continuously depend on the initial conditions of the problem. The purpose of this article is to determine sufficient conditions for the existence of a solution to the Sturm-Liouville equation with a weighty coefficient and smoothness of the solution. In the course of achieving the goal, the given differential equation was studied by the method of operators in a functional space. The problem was considered in a Banach space, and the properties of functional spaces were used. It is known that many questions of quantum mechanics are reduced to problems in which the emission of electromagnetic waves in a Hilbert space is determined in the special case by the presence of inverse operators of singular differential operators and separability. One of these operators is the Sturm-Liouville operator with weight at the highest derivative. In this paper, the named operator is investigated by the methods of functional analysis. Sufficient conditions for the existence of a solution and separability of an operator in a Banach space are found.