The links, presently used in most of the industrial robots-manipulators, are rigid and heavy, which allows us to neglect deformations during the working operations. Such kind of a robot has certain drawbacks: low speed; high energy consumption; low payload-to-weight ratio, etc. Application of the robots with lightweight links gives a number of advantages. Namely: better payload-to-weight ratio; higher speed of movement; increased safety; lower energy consumption; bigger working space with the use of lengthened links; and lower cost. However, such type of a manipulator has an essential drawback - flexibility of a link, which complicates its mathematical model. The flexible single-link manipulator considered in the article as a control object has been attractive for the control specialists for a long time. A regular approach to the design control systems begins with development of a mathematical model of a plant, which describes its dynamic properties in the best way. However, an analysis of the numerous articles and monographs on the problem revealed noticeable distinctions in the used mathematical models, while the initial data concerning the design and operation of the manipulator were the same. Especially this concerns the boundary conditions, the orthogonality conditions and expressions for the flexible mode shapes. In this paper all the formulas for the free and forced motions of the flexible manipulator were received only on the basis of the Hamilton's principle and assumed mode method, namely: the governing equation of motion - Euler-Bernoulli equations, boundary conditions, orthogonality conditions, expressions for the mode shapes and also finite dimensional approximation of the model in the state space. The development of the appropriate formulas is described in every detail.