@article{DAVENPORT1986237,
    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 u1,..., log um), 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.},
    Author = {Davenport, J.H. and Singer, M.F.},
    File = {Elementary and Liouvillian Solutions of Linear Differential Equations - 1-s2.0-S0747717186800268-main.pdf},
    ISSN = {0747-7171},
    Journal = {Journal of Symbolic Computation},
    Number = {3},
    Pages = {237-260},
    Title = {Elementary and Liouvillian Solutions of Linear Differential Equations},
    URL = {https://www.sciencedirect.com/science/article/pii/S0747717186800268},
    Volume = {2},
    Year = {1986},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0747717186800268},
    bdsk-url-2 = {https://doi.org/10.1016/S0747-7171(86)80026-8},
    date-added = {2022-12-20 11:43:02 +0100},
    date-modified = {2022-12-20 11:43:02 +0100},
    doi = {10.1016/S0747-7171(86)80026-8}
}

@article{DAVENPORT1986237, 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 u1,..., log um), 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.}, Author = {Davenport, J.H. and Singer, M.F.}, File = {Elementary and Liouvillian Solutions of Linear Differential Equations - 1-s2.0-S0747717186800268-main.pdf}, ISSN = {0747-7171}, Journal = {Journal of Symbolic Computation}, Number = {3}, Pages = {237-260}, Title = {Elementary and Liouvillian Solutions of Linear Differential Equations}, URL = {https://www.sciencedirect.com/science/article/pii/S0747717186800268}, Volume = {2}, Year = {1986}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0747717186800268}, bdsk-url-2 = {https://doi.org/10.1016/S0747-7171(86)80026-8}, date-added = {2022-12-20 11:43:02 +0100}, date-modified = {2022-12-20 11:43:02 +0100}, doi = {10.1016/S0747-7171(86)80026-8} }