Elementary and Liouvillian Solutions of Linear Differential Equations

Abstract

Let L(y) = b be a linear differential equation with coefficients in a differential field k , of characteristic 0, We show that if L(y) = b has a non-zero solution Liouvillian over k , then either L(y) = 0 has a non-zero solution u such that u'/u is algebraic over k , or L(y) = b has a solution in k . If L(y) = b has a non-zero solution elementary over k , then either L(y) = 0 has a non-zero solution algebraic over k, or L(y) = b has a solution in k . This latter fact is a consequence of the fact that if L(y) = b has a solution elementary over k , then it has a solution of the form P(log u 1 ,..., log u m ), where P is a polynomial with coefficients algebraic over k whose degree is at most equal to the order of L(y) , and the u ; are algebraic over k . Algorithmic considerations are also discussed.