On Liouvillian integrability of the first-order polynomial ordinary differential equations (Q448282)
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scientific article; zbMATH DE number 6074428
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On Liouvillian integrability of the first-order polynomial ordinary differential equations |
scientific article; zbMATH DE number 6074428 |
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On Liouvillian integrability of the first-order polynomial ordinary differential equations (English)
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30 August 2012
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Liouvillian integrability
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invariant algebraic curve
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Riccati differential equation
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Abel differential equation
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The authors prove the following result:NEWLINENEWLINE Theorem. If a complex differential equation of the form NEWLINE\[NEWLINE{dy\over dx}= a_0(x)+ a_1(x) y+\cdots+ a_n(x) y^n,NEWLINE\]NEWLINE where \(a_i(x)\), \(i= 0,\dots, n\), are polynomials in \(x\), \(a_n(x)\neq 0\), \(n\geq 2\), has a Liouvillian first integral, then it has a finite invariant algebraic curve.
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