This seems to work to me. I reposted it on MathOverflow (giving you credit of course) to see if they could verify it (as that is where the question originally came from), although I would also appreciate it if someone here could check this or provide an alternate answer. Thanks.
Given a differential field F and a linear algebraic group G over the constant field C of F, find a Picard-Vessiot extension of E of F with G(E/F)=G:
This isn't homework, just something I saw in a book that I was curious about. The author says that this can be shown but doesn't illustrate how...