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Liouvillian First Integrals of Differential Equations [PDF]

Transactions of the American Mathematical Society, 1988
Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this ...
M. Singer
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An Efficient Method for Computing Liouvillian First Integrals of Planar Polynomial Vector Fields [PDF]

Journal of Differential Equations, 2020
Here we present an efficient method to compute Darboux polynomials for polynomial vector fields in the plane. This approach is restricetd to polynomial vector fields presenting a Liouvillian first integral (or, equivalently, to rational first order ...
L. Duarte, L. D. Mota
semanticscholar   +4 more sources

LIOUVILLIAN FIRST INTEGRALS OF HOMOGENEOUS POLYNOMIAL 3-DIMENSIONAL VECTOR FIELDS [PDF]

Colloquium Mathematicum, 1996
Summary: Given a three-dimensional vector field \(V\) with coordinates \(V_x\), \(V_y\) and \(V_z\) that are homogeneous polynomials in the ring \(k[x, y, z]\), we give a necessary and sufficient condition for the existence of a Liouvillian first integral of \(V\) which is homogeneous of degree 0.
J. M. Ollagnier
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Differential Algebra and Liouvillian first integrals of foliations

Journal of Pure and Applied Algebra, 2011
Let \((k,\Delta)\) be a differential field, where \(k\) is a field and \(\Delta\) is a set of derivations of \(k\). A Liouvillian extension of \((k,\Delta)\) is a differential extension \((K,\tilde{\Delta})\) of \( (k,\Delta)\) for which there is a chain of differential extensions \(k=k_{0}\subset k_{1}\subset\dots\subset k_{m}=K\) such that \( k_{i+1}/
B. Scárdua
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On the nonexistence of Liouvillian first integrals for generalized Liénard polynomial differential systems [PDF]

Journal of Nonlinear Mathematical Physics, 2013
We consider generalized Liénard polynomial differential systems of the form ẋ = y, ẏ = -g(x) - f (x) y, with f (x) and g(x) two polynomials satisfying deg(g) ≤ deg(f). In their work, Llibre and Valls have shown that, except in some particular cases, such
Guillaume Chèze, T. Cluzeau
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Liouvillian first integrals of differential equations

Banach Center Publications, 2011
In this paper we generalize to any dimension and codimension some theorems about existence of Liouvillian solutions or first integrals proved by M. Singer in Liouvillian first integrals of differential equations (1992) for first order differential equations.
G. Casale
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Liouvillian first integrals of quadratic–linear polynomial differential systems

Journal of Mathematical Analysis and Applications, 2011
For a large class of quadratic–linear polynomial differential systems with a unique singular point at the origin having non-zero eigenvalues, we classify the ones which have a Liouvillian first integral, and we provide the explicit expression of them.
J. Llibre, C. Valls
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A note on Liouvillian first integrals and invariant algebraic curves

Applied Mathematics Letters, 2013
In this paper we study the existence and non-existence of finite invariant algebraic curves for complex planar polynomial differential system having a Liouvillian first integral.
Jaume Giné, M. Grau, J. Llibre
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Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms [PDF]

Symmetry, Integrability and Geometry: Methods and Applications, 2006
It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a product of ...
Yuri N. Kosovtsov
doaj   +5 more sources

On Liouvillian integrability of the first–order polynomial ordinary differential equations

Journal of Mathematical Analysis and Applications, 2012
The authors prove the following result: Theorem. If a complex differential equation of the form \[ {dy\over dx}= a_0(x)+ a_1(x) y+\cdots+ a_n(x) y^n, \] 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.
J. Giné, J. Llibre
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