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

摘要

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.The method is illustrated by examples of wide classes of Liénard I-type equations. These classes contain but not limited to generalized force-free Duffing-Van der Pol oscillator equations. Analytical solutions are explicitly provided for both oscillatory and nonoscillatory types of equations.