Asymptotically close solutions to an ordinary differential equation

We consider an ordinary differential equation of a very general form. Let according to Preprint no. 9/2003 we have found a power-logarithmic expansion of its solution with the power asymptotics. Here we show how by methods of Power Geometry one can compute all power and exponential additions to the expansion, corresponding to such solutions to the equation that are close to the found one. This allows to find some exponentially small effects by means of Power Geometry. We give examples of the calculations. The main attention is given to explanations of the computational algorithms.