You are asking about a very classical problem. The Picard-Vessiot theory was developed to show that, in a certain well-defined sense, there is no `closed form' solution to problems of this kind. You should take a look at the books by Kolchin and Ritt on differential algebra. For a start on the basic ideas, have a look at this paper by Hubbard and Lundell:

http://www.math.cornell.edu/~hubbard/diffalg1.pdf