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:
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