Given two ordinary differential equations of order n and m, when is there a relation between two generic tuples of solutions? To answer this question, we would like to bound the length of the tuples one has to check by a quantity only depending on n and m.

In this talk, I will show how this problem is related, via model-theoretic Galois theory, to generically transitive differential group actions. In particular, I will present a proof of a variant of the Borovik-Cherlin conjecture for differentially closed fields and show how it can be used to obtain a strong bound on one of the tuples, while giving up all control on the length of the other.