The stability of helicopter behaviour during flight depends on many factors. One of them is directly related to the non-stationary elastic bending vibrations of the rotor blades. Air pressure on the blade surface causes bending deformations in this element. Since the movement of the helicopter rotor is periodic, the non-stationary load on the blade varies according to a harmonic law. As a result, we have a variable pressure and a change in lift. All of the above leads to a change in aerodynamic parameters, which causes helicopter motion instability. In this paper, the study of the blade's bending vibrations is based on its modelling as a thin rectangular plate. The available experimental data from photographs recorded the movement of the helicopter blade in the form of oscillations similar to those of a cantilevered plate. It is known that helicopter blades are made quite rigid due to the existing stiffeners - nerves. Therefore, in this work, the Kirchof-Love hypothesis and the Lagrange-Sophie Germain equation are used to simulate the non-stationary bending vibrations of a helicopter blade as a thin plate. A review of existing works has shown that many studies have been dedicated to the study of bending vibrations of a rectangular plate described by the Lagrange-Sophie Germain equation. However, there is no exact solution to the problem of free vibrations of a plate in the case of its cantilever mounting. There are only approximate solutions using the Ritz method. In this paper, an attempt was made to solve analytically the problem of free oscillations of small amplitude of a thin plate under classical boundary conditions of cantilevered plate fixation by two different methods. As it turned out, the analytical solution obtained by the Levy method is not unique. The system of equations for determining the coefficients of the general solution has a rank less than the number of unknowns. This indicates the dependence of the eigenvalues of the problem, i.e. the coupling of oscillations, which was previously found by Levy under other boundary conditions. Since the goal of this paper is to find a single solution, Part 2 of this paper presents an example of refined boundary conditions that allowed us to find a single solution to the problem.