Consider a rather general system of ordinary differential equations (ODEs). Assume that its truncated system has a solution in the form of powers of the independent variable multiplied by series in powers of its multiple logarithms, In the absence of critical values, it is shown that this nonpower asymptotic form of the solution to the original system can be extended to an expansion of the solution to the original equation. As a result, we obtain series in powers of the independent variable whose coefficients are series in powers of its multiple logarithms. Examples of such computations are presented. Major emphasis is placed on explanations of the computational algorithms.
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