29 pages Following an initial referee's report on this version, the authors found a mistake in one of the results about finding strongly normal extensions with additional properties. Correcting this mistake led to a substantial rewriting of the paper, with additional hypotheses, and the revised version was accepted by International Math. Research Notices

We give model theoretic accounts and proofs of the existence and uniqueness of differential Galois extensions with no new constants, for logarithmic differential equations over a differential field K, when the field C of constants of K is not necessarily algebraically closed, under a variety of assumptions on C and K.