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On the Jacobi last multipliers and Lagrangians for a family of Liénard-type equations

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  • Sinelshchikov, Dmitry I.
  • Kudryashov, Nikolay A.

Abstract

We study a family of the Liénard-type equations, which can be transformed via the generalized Sundman transformations into a particular case of Painlevé–Gambier equation XXVII. We show that this equation of the Painlevé–Gambier type admits an autonomous Lagrangian, Jacobi last multiplier and first integral. As a consequence, we obtain that the corresponding family of Liénard-type equations also admits a time-independent Lagrangian, Jacobi last multiplier and first integral. We also construct the general analytical and singular solutions for members of this family of Liénard-type equations by virtue of the generalized Sundman transformations. To demonstrate applications of our results we consider several examples of the Liénard-type equations, with a generalization of the modified Emden equation among them, and construct their autonomous Lagrangians, Jacobi last multipliers and first integral as well as their general analytical solutions.

Suggested Citation

  • Sinelshchikov, Dmitry I. & Kudryashov, Nikolay A., 2017. "On the Jacobi last multipliers and Lagrangians for a family of Liénard-type equations," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 257-264.
    • Handle: RePEc:eee:apmaco:v:307:y:2017:i:c:p:257-264
    DOI: 10.1016/j.amc.2017.03.010
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    Cited by:

    1. Polyanin, Andrei D. & Shingareva, Inna K., 2018. "Nonlinear problems with blow-up solutions: Numerical integration based on differential and nonlocal transformations, and differential constraints," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 107-137.
    2. Sinelshchikov, Dmitry I., 2020. "On linearizability via nonlocal transformations and first integrals for second-order ordinary differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    3. Ruiz, A. & Muriel, C., 2018. "On the integrability of Liénard I-type equations via λ-symmetries and solvable structures," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 888-898.

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