Transition from laminar to turbulent flow has been a primary concern on the study of a pipe flow. This study provides the analytical solutions of the axisymmetric Navier-Stokes equations for an arbitrary-accelerating laminar flow with a given cross-sectional mean velocity in a circular pipe. The cross-sectional mean velocity is imposed as cubic polynomial with respect to time. The exact solutions obtained are thought to be useful to verify the accuracy of the corresponding experimental results for investigating the transition process to turbulence and, to control the cross-sectional mean velocity in the experiment. Unfortunately, the exact solutions involve the infinite series with respect to the time and the zeros of the second-order usual Bessel function of the first kind. To avoid a difficulty of computing the infinite series for a very small time, this study investigates the asymptotic behavior for the early stage of motion. In addition, a singular perturbation approach is presented in Appendix and the truncation error in computing the infinite series is investigated. As a result, a radial velocity distribution, a pressure gradient and a shear stress are given for four kinds of typical acceleration patterns and compared with the preceding experimental results on the radial velocity distribution. Thickness of the boundary-layer, the pressure gradient and the shear stress are also calculated at a transition point to turbulence using the previous experimentally measured values of a transition time.