Counterexamples to the assumption on the possibility of prolongation of truncated solutions of a truncated LODE

Previously, the authors proposed algorithms making it possible to find exponential-logarithmic solutions of linear ordinary differential equations with coefficients in the form of power series in which only the initial terms are known. The solution includes a finite number of power series, and the maximum possible number of their terms is calculated. Now, these algorithms are supplemented with the option to confirm the impossibility of obtaining a larger number of terms in the series without using additional information about the given equation a counterexample is constructed to the assumption that it is possible to obtain uniquely defined additional terms. In previous papers, the authors proposed such confirmations for the cases of Laurent and regular solutions.