In this paper, Lagrange’s equations along with the Ritz method are used to obtain the equation of motion for a flexible, slender cylinder subjected to axial flow. The cylinder is supported only by a translational and a rotational spring at the upstream end, and at the free end, it is terminated by a tapering end-piece. The equation of motion is solved numerically for a system in which the translational spring is infinitely stiff, thus acting as a pin, while the stiffness of the rotational spring is generally non-zero. The dynamics of such a system with the rotational spring of an average stiffness is described briefly. Moreover, the effects of the length of the cylinder and the shape of the end-piece on the critical flow velocities and the modal shapes of the unstable modes are investigated.