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On the integrability of Liénard I-type equations via λ-symmetries and solvable structures

Author

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  • Ruiz, A.
  • Muriel, C.

Abstract

For Liénard I-type equations it is proved the existence of a family of λ−symmetries such that any of them lets the computation by quadratures of a time-dependent first integral of the equation. This is achieved by using a solvable structure constructed out of the λ−symmetry and one Lie point symmetry. The first integral obtained by quadratures and the first integral associated to the Lie symmetry generator are always functionally independent and they can be therefore used to integrate completely the Liénard I-type equation.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:339:y:2018:i:c:p:888-898
    DOI: 10.1016/j.amc.2018.07.056
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    References listed on IDEAS

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    1. 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.
    2. Najeeb Alam Khan & Muhammad Jamil & Syed Anwar Ali & Nadeem Alam Khan, 2011. "Solutions of the Force-Free Duffing-van der Pol Oscillator Equation," International Journal of Differential Equations, Hindawi, vol. 2011, pages 1-9, October.
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    Cited by:

    1. 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).
    2. Sinelshchikov, Dmitry I., 2021. "Nonlocal deformations of autonomous invariant curves for Liénard equations with quadratic damping," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).

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