Integrating Factors and λ—Symmetries

Muriel, C.; Romero, J. L.

doi:10.2991/jnmp.2008.15.s3.29

Cited by 41 publications

(58 citation statements)

References 8 publications

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“…We have already mentioned that there are equations in A that lack Lie point symmetries. Recently, several relationships between first integrals and λ-symmetries have been derived ( [14,15]). In Sec.…”

Section: A(t X)ẋ + B(t X) (13)mentioning

confidence: 99%

Second-Order Ordinary Differential Equations and First Integrals of The Form A(t, x) ẋ + B(t, x)

Muriel¹,

Romero²

2021

JNMP

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We characterize the equations in the class A of the second-order ordinary differential equations x = M (t, x,ẋ) which have first integrals of the form A(t, x)ẋ + B(t, x). We give an intrinsic characterization of the equations in A and an algorithm to calculate explicitly such first integrals. Although A includes equations that lack Lie point symmetries, the equations in A do admit λ-symmetries of a certain form and can be characterized by the existence of such λ-symmetries. The equations in a well-defined subclass of A can completely be integrated by using two independent first integrals of the form A(t, x)ẋ + B(t, x). The methods are applied to several relevant families of equations.

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Section: A(t X)ẋ + B(t X) (13)mentioning

confidence: 99%

Second-Order Ordinary Differential Equations and First Integrals of The Form A(t, x) ẋ + B(t, x)

Muriel¹,

Romero²

2021

JNMP

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“…In the case of second-order ODEs, the λ-symmetry is nothing but the null form with a negative sign [12]. This one-to-one correspondence came from the result that the S-determining equation in the Prelle-Singer procedure differs only by a negative sign from that of the λ-symmetry determining equation [1].…”

Section: Interconnectionsmentioning

confidence: 99%

“…In the case of second-order ODEs, the λ-symmetry is nothing but the null form with a negative sign. This one-to-one correspondence came from the result that the S-determining equation in the PrelleSinger procedure differs only by a negative sign from that of the λ-symmetry determining equation [12]. In other words, there is a one-to-one correspondence between the λ-symmetries and the null form S. However, in the case of third-order ODEs, we have two null forms (S and U) which have to be connected to the single function λ.…”

Section: (F) Comparison Between Interconnections For Second-and Thirdmentioning

confidence: 99%

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Interconnections between various analytic approaches applicable to third-order nonlinear differential equations

et al. 2015

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We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ODEs. We establish an important interconnection between the extended Prelle-Singer procedure and λ-symmetries approach applicable to third-order ODEs to bring out the various linkages associated with these different techniques. By establishing this interconnection we demonstrate that given any one of the quantities as a starting point in the family consisting of Jacobi last multipliers, Darboux polynomials, Lie point symmetries, adjointsymmetries, λ-symmetries, integrating factors and null forms one can derive the rest of the quantities in this family in a straightforward and unambiguous manner. We also illustrate our findings with three specific examples.

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“…This process of reduction of order lets us derive integrating factors of a given equation from integrating factors of the reduced equations ( [8]). For simplicity we explain here this procedure for second order equations:…”

Section: Conserved Forms Derived From λ−Symmetriesmentioning

confidence: 99%

Conserved Forms derived from Symmetries

Muriel

Romero

2008

Proc Appl Math and Mech

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For first order scalar ordinary differential equations, a well-known result of Sophus Lie states that a Lie point symmetry can be used to construct an integrating factor and conversely.However, there exist higher order equations without Lie point symmetries that admit integrating factors or that are exact. We present a method based on λ−symmetries to calculate integrating factors. An example of a second order equation without Lie point symmetries illustrates how the method works in practice and how the computations that appear in other methods may be simplified. is a function µ(x, u (n−1) ) such that the equation µ · ∆ = 0 is an exact equation:Function G(x, u (n−1) ) in (2) will be called a first integral of the equation (1) and For first order equations Sophus Lie proved the equivalence between integrating factors and Lie point symmetries ([6]). For higher order equations we can not expect a similar equivalence. There are equations of order n > 1 without Lie point symmetries that admit integrating factors or that are exact ([10], pag.182). For this reason, it has been tried to connect integrating factors with other types of symmetries, such as nonlocal symmetries ([5]). The main problem is that, as far as we know, there is no method to calculate nonlocal symmetries and no general method valid for arbitrary ODEs has been presented.Our contribution to this problem proves that integrating factors are equivalent to the λ−symmetries of the equation ([8]). At difference of nonlocal symmetries, λ−symmetries can be calculated by a well-defined algorithm ([7]). In particular, a vector field v = ∂ u is a λ−symmetry of equation (1) for any particular solution λ = λ(x, u (n−1) ) to the partial differential equationλ−Symmetries have associated an algorithm to reduce the order of the equation and this method unifies most of the known processes of order reduction, even those not derived by the existence of Lie point symmetries. This process of reduction of order lets us derive integrating factors of a given equation from integrating factors of the reduced equations ([8]). For simplicity we explain here this procedure for second order equations:In this case, determining equation (

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Integrating Factors and λ—Symmetries

Cited by 41 publications

References 8 publications

Second-Order Ordinary Differential Equations and First Integrals of The Form A(t, x) ẋ + B(t, x)

Second-Order Ordinary Differential Equations and First Integrals of The Form A(t, x) ẋ + B(t, x)

Interconnections between various analytic approaches applicable to third-order nonlinear differential equations

Conserved Forms derived from Symmetries

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