We study self-similar hydrodynamics of spherical symmetry using a general polytropic ( GP) equation of state and derive the GP dynamic Lane-Emden equation ( LEE) with a radial inertial force. In reference to Lou & Cao, we solve the GP dynamic LEE for both polytropic index gamma = 1 + 1/n and the isothermal case n -> +infinity; our formalism is more general than the conventional polytropic model with n = 3 or gamma = 4/3 of Goldreich & Weber. For proper boundary conditions, we obtain an exact constant solution for arbitrary n and analytic variable solutions for n = 0 and n = 1, respectively. Series expansion solutions are derived near the origin with the explicit recursion formulae for the series coefficients for both the GP and isothermal cases. By extensive numerical explorations, we find that there is no zero density at a finite radius for n >= 5. For 0 <= n < 5, we adjust the inertial force parameter c and find the range of c > 0 for monotonically decreasing density from the origin and vanishing at a finite radius for c being less than a critical value C-cr. As astrophysical applications, we invoke our solutions of the GP dynamic LEE with central finite boundary conditions to fit the molecular cloud core Barnard 68 in contrast to the static isothermal Bonnor-Ebert sphere by Alves et al. Our GP dynamic model fits appear to be sensibly consistent with several more observations and diagnostics for density, temperature and gas pressure profiles.