Complicated expansions of solutions to a system of ordinary differential equations

The examples of the computing complicated expansions of solutions to system of ordinary differential equations, in the form of the independent variables, are presented. The functions depending on an independent variable are expanded in series in the decreasing powers of logarithm of that independent variable. The single differential equation is computed with the help of truncated system, satisfying certain constraints in the form of a matrix. The results show that if truncated solution has no critical values, then it is associated with unique expansion. The critical values of nonpower asymptotic forms are computed and are found to be the coefficients of relational functions of certain parameters. The nonpower asymptotic form of ordinary differential equations can be extended to the complicated expansion.