On complicated expansions of solutions to ODE

We consider an ordinary differential equation of a very general form. Let its truncated equation corresponding to a vertex or to a nonhorisontal edge of the polygon of the initial equation have a solution containing the logarithm of the independent variable. We demonstrate that, under a very weak restriction, the nonpower asymptotic form of solutions to the initial equation can be continued into an asymptotic expansion of the solutions. The expansion is in powers of the independent variable with coefficients that are series in decreasing powers of its logarithm. We demonstrate algorithms of such computations. We also give 6 examples, 4 of them are related to to Painlevè equations.