We consider linear ordinary differential systems over a differential field of characteristic 0. We prove that testing unimodularity and computing the dimension of the solution space of an arbitrary system can be done algorithmically if and only if the zero testing problem in the ground differential field is algorithmically decidable. Moreover, we consider full-rank systems whose coefficients are computable power series and we show that, despite the fact that such a system has a basis of formal exponential-logarithmic solutions involving only computable series, there is no algorithm to construct such a basis.
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