Isochronous waveforms of Liénard equations via commutative factorization

• Isochronous trigonometric solutions of Liénard equations by commutative factorization. • Such solutions are of Bernoulli class, i.e., of corresponding Bernoulli nonlinear equations. • Phase portraits are presented for three important examples. • Theorem stating the general form of the Liénard equations solvable in this way. Isochronous waveform solutions of homogeneous Liénard equations are obtained by a modification of the nonlinear factorization method of Rosu and Cornejo-Pérez. The scheme is based on the assumption that the intermediate function Φ that can be introduced in this factorization method depends on both the dependent and independent variables of the nonlinear equation. The method is applied to three cases, a noted cubic anharmonic oscillator, a Liénard-reduced form of the Sharma-Tasso-Olver evolution equation, and the cubic-quintic Wilson’s Liénard equation. All these cases are written in a commutative factored form that allows to obtain the general solutions as solutions of a certain type of Bernoulli differential equation. A theorem is also given asserting the general form of the Liénard equation, i.e., for given polynomial degree n of its coefficients, which can be solved by this method. The conditions under which these equations can be also approached by non-local transformations are established.