Application of the Generalised Sundman Transformation to the Linearisation of Two Second-Order Ordinary Differential Equations

Moyo, Sibusiso; Meleshko, Sergey V.

doi:10.1142/s1402925111001386

Cited by 22 publications

(17 citation statements)

References 14 publications

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“…is called a generalised Sundman transformation [2][3][4]12]. Several properties and characterizations of the class A of the second-order ODEs that can be transformed into the linear equation U XX = 0 by means of (2.17) were derived in [2,16,17].…”

Section: Linearisation By Generalised Sundman Transformationsmentioning

confidence: 99%

“…Remarkably, none of the equations under study are linearisable by point transformations. Nevertheless, the equations in the family can be linearised by generalised Sundman transformations [2][3][4]12], which can be constructed from the coefficients of the associated first integral [16,17]. The relationships between generalised Sundman transformations and Sundman symmetries [3], or between λ −symmetries and telescopic vector fields [21], exponential vector fields [20], semi-classical nonlocal symmetries [1,5], can be used to derive these different types of symmetries of the equations under study (see [15] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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Exact solutions and Riccati-type first integrals

Mendoza

Muriel

2021

JNMP

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The λ −symmetry approach is applied to a family of second-order ODEs whose algebra of Lie point symmetries is insufficient to integrate them. The general solution and two functionally independent first integrals of a subclass of the studied equations can be expressed in terms of a fundamental set of solutions of a related second-order linear ordinary differential equation. Remarkably, no equation in the family passes the Lie's test of linearisation, although all of them can be linearised by generalised Sundman transformations. Several examples, including equations lacking Lie point symmetries, illustrate the presented procedure.

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Section: Linearisation By Generalised Sundman Transformationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exact solutions and Riccati-type first integrals

Mendoza

Muriel

2021

JNMP

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“…where R = f yy = 0. Therefore, we have found the necessary and sufficient conditions for equation (1) to possess a first integral in the form (17). These conditions splits into six separate cases: generic case (23) and special cases (25), (27) and (28), (29), (31), (32) and (34).…”

Section: The Functions a And B Are Defined By The Relationsmentioning

confidence: 99%

“…Let us consider equation (2) and obtain the necessary and sufficient conditions for the existence of first integral (17). Overdetermined system of equations for the parameters of this first integral is completely the same as in the case of equation (1).…”

Section: The Classical Liénard Equationmentioning

confidence: 99%

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Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

Sinelshchikov

Gaiur

Kudryashov

2019

Journal of Mathematical Analysis and Applications

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We consider a family of non-autonomous second-order differential equations, which generalizes the Liénard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold-Liouville integrability of Hamiltonian systems. We illustrate our results by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation.

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Lie group approach for constructing all reciprocal transformations: The two‐dimensional stationary gas dynamics equations

Siriwat

Meleshko

2022

Math Methods in App Sciences

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Recently, an infinitesimal approach for finding reciprocal transformations has been proposed. The method uses the group analysis approach and consists of similar steps as for finding an equivalence group of transformations. The new method provides a systematic tool for finding classes of reciprocal transformations (group of reciprocal transformations). Similar to the classical group analysis, this approach can be also applied for finding all reciprocal transformations (not only composing a group) of the equations under study. The present paper provides this algorithm. As an illustration, the method is applied to the two-dimensional stationary gas dynamics equations. Equivalence group, group of reciprocal transformations, and completeness of all discrete reciprocal transformations are presented in the paper. The results are stated in form of a theorem. KEYWORDSequivalence group of transformations, Lie group of transformations, reciprocal transformations MSC CLASSIFICATION 76M60, 35R10 * See also references therein.

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Application of the Generalised Sundman Transformation to the Linearisation of Two Second-Order Ordinary Differential Equations

Cited by 22 publications

References 14 publications

Exact solutions and Riccati-type first integrals

Exact solutions and Riccati-type first integrals

Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

Lie group approach for constructing all reciprocal transformations: The two‐dimensional stationary gas dynamics equations

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