By means of Power Geometry, shortly presented in §1, in the generic case we compute all power expansions of solutions to the second Painleve' equation at points z = 0, z = ∞ (§2) and z = z₀ ≠ 0 (§3). Analogously for a = 0 we compute all power expansions of solutions and of logarithm of solutions (§4). We have found new fine properties of some of these expansions.

Mathematical problems and theory of numerical methods