The problem of forced oscillations of an open section of a gas pipeline during the cleaning piston passage belongs to the type of problems of forced oscillations of one-dimensional elastic objects under the influence of a moving inertial load on them. Currently, there are two ways to solve such problems. The first way is related to the integration of the partial differential equation and the solution of such problems is a superposition of eigenoscillations and accompanying oscillations. The second way does not involve the integration of the partial differential equation. Methods of generalized coordinates, generalized displacements and various numerical methods belong to the second type of solving. None of the mentioned methods is simple. Therefore, the authors suggest the method, in which the first mathematical model provides the determination of forced oscillations of the gas pipeline section during the passage of the cleaning piston without taking into account its inertial load on the gas pipeline. In future, on the basis of the first model it is planned to develop the second mathematical model which will provide an approximate determination of the deflections of the pipeline axis, taking into account the inertial load of the piston on the pipeline. The purpose of this article is to obtain a solution to the problem of forced oscillations of the pipeline section during the passage of the cleaning piston without taking into account the inertial forces on the pipeline. The problem is solved by partial differential equation, Fourier method is applied. The right side of the nonhomogeneous differential equation is decomposed into an infinite series, which is the sum of the productions of the eigenfunctions of the pipeline section free oscillations and the unknown function of time. After finding out this function, the authors determine the unknown time function in the Fourier method and hence the solution of the problem in the form of an infinite series, the summands of which lessen rapidly. The authors calculate the deflections of the pipeline axis along the entire section of the gas pipeline for different points of time, as well as deflections of individual sections changing in time and moments of deflection