Liouvillian solutions of first order nonlinear differential equations. (English) Zbl 1418.34004
Summary: Let \(k\) be a differential field of characteristic zero and \(E\) be a Liouvillian extension of \(k\). For any differential subfield \(K\) intermediate to \(E\) and \(k\), we prove that there is an element in the set \(K-k\) satisfying a linear homogeneous differential equation over \(k\). We apply our results to study Liouvillian solutions of first order nonlinear differential equations and provide generalisations and new proofs for several results of M. F. Singer [J. Symb. Comput. 10, No. 1, 59–94 (1990; Zbl 0727.12011)] and M. Rosenlicht [Proc. Am. Math. Soc. 37, 369–373 (1973; Zbl 0253.12105)] on this topic.
MSC:
12H20 | Abstract differential equations |
12H05 | Differential algebra |
34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
differential field; linear homogeneous differential equation; Liouvillian solutions; first order nonlinear differential equationsReferences:
[1] | Chambert-Loir, A., A Field Guide to Algebra (2005), Springer: Springer New York · Zbl 1155.12001 |
[2] | Kolchin, E. R., Differential Algebra and Algebraic Groups (1973), Academic Press: Academic Press New York · Zbl 0264.12102 |
[3] | van der Put, M.; Singer, M. F., Galois Theory of Linear Differential Equations, Grundlehren Math. Wiss., vol. 328 (2003), Springer: Springer Heidelberg · Zbl 1036.12008 |
[4] | Rosenlicht, M., An analogue of l’Hospital’s rule, Proc. Am. Math. Soc., 37, 2, 369-373 (1973) · Zbl 0253.12105 |
[5] | Singer, M., Elementary solutions of differential equations, Pac. J. Math., 59, 2, 535-547 (1975) · Zbl 0315.12103 |
[6] | Singer, M., Liouvillian solutions of \(n\) th order homogeneous linear differential equations, Am. J. Math., 103, 4, 661-682 (1981) · Zbl 0477.12026 |
[7] | Singer, M., Formal solutions of differential equations, J. Symb. Comput., 10, 59-94 (1990) · Zbl 0727.12011 |
[8] | Srinivasan, Varadharaj R., Iterated antiderivative extensions, J. Algebra, 324, 8, 2042-2051 (2010) · Zbl 1226.12003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.