Integrating Factors and First Integrals for Liénard Type and Frequency-Damped Oscillators

We consider Liénard type and frequency-damped oscillator equations. Integrating factors and the associated first integrals are derived from the method to compute λ-symmetries and the associated reduction algorithm. The knowledge of a λ-symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. We will compare our results with those gained by the other methods.


Introduction
Integrating factors and first integrals are powerful tools in the study of ordinary differential equations ODEs .In this area, we observe plenty of studies.Some of them are listed in 1-8 .reduction algorithm.In addition, the knowledge of a λ-symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of PDEs.
The main purpose of this paper is to study integrating factors and first integrals of two important oscillation equations which are Liénard type and frequency-damped oscillation equations by the λ-symmetry methods.An interesting feature of our study is that the method of 10 does not produce integrating factors for Liénard type oscillation equation; however, λ-symmetry methods yield integrating factor and first integral.Moreover, we obtain new integrating factor by λ-symmetry methods different from that which is previously derived by 10 for frequency-damped oscillation equation.
The paper is structured thusly.In Section 2, we present the necessary preliminaries.This section is devoted to integrating factors and the associated first integrals which are derived from the method to compute λ-symmetries and the associated reduction algorithm.In addition, we introduce Anco and Bluman's method 10 .
In Section 3, we study the λ-symmetries, integrating factors, first integrals for Liénard type, and frequency-damped oscillator equations.In addition, we make comparisons between two methods which are pointed out briefly above.The final section includes some conclusions.

Necessary Preliminaries
We first present notation to be used and recall the definitions and theorems that appear in 10, 11 .
Consider an nth-order ordinary differential equation denotes the corresponding k-jet space, and the elements of M k are denoted by t, x k t, x, ẋ, . . ., k x .Suppose that 2.1 admits an integrating factor, μ t, x k , for some k such that 0 ≤ k ≤ n − 1.The multiplication by μ converts the left-hand side of 2.1 into the total derivative of some function Δ t, x n−1 , The exact equation D t Δ t, x n−1 0 admits a λ-symmetry, and the trivial reduction of order Δ t, x n−1 C, C ∈ Ê, appears as a consequence of the reduction algorithm associated to that λ-symmetry 13 then the vector field v ∂ x is a λ-symmetry of 2.1 where F is an analytic function of its arguments in some open subset M n−1 , and it appears when 2.1 is locally written in explicit form.
Theorem 2.1 see 12 .Assume that 2.1 is an nth-order ordinary differential equation that admits an integrating factor μ such that μ| Δ 0 / 0. If λ is any particular solution of 2.3 , then the vector field v ∂ x is a λ-symmetry of 2.1 .
An algorithm to obtain an integrating factor or first integral of a given ODE once a λ-symmetry is known has been derived in 12 .For second-order equations, the method reads as follows.
For n 2, the corresponding second-order ODEs can be written in explicit form as ẍ F t, x, ẋ .

2.4
We denote by A ∂ t ẋ∂ x F t, x, ẋ ∂ ẋ the vector field associated with 2.4 .In terms of A, a first integral of 2.4 is any function I t, x, ẋ such that A I 0. Let v be a λ-symmetry of 2.4 , Then w t, x, ẋ is a first-order invariant of v λ, 1 , that is, any particular solution of the equation The reduction process associated to the λ-symmetry v gives a first-order reduced equation of the form Δ R t, x, ẋ 0, the general solution of which is implicitly given by an equation of the form G t, w is an integrating factor of 2.4 .
For second-order ODEs, there is a correspondence between λ-symmetries and first integrals in the sense of Theorem 2 in 11 .
4 for some function λ t, x, ẋ , then there exists a first integral I t, x, ẋ of 2.4 such that v λ, 1 I 0. Theorem 2.2 works when v ∂ x is a λ-symmetry.If I is a first integral of 2.4 , then μ I ẋ is an integrating factor of 2.4 and −μφ I x ẋI x .If I is also a first integral of v λ, 1 for some function λ t, x, ẋ , then I x −λI ẋ −λμ and the system is compatible; that is, when λ t, x, ẋ is such that v ∂ x is λ-symmetry, we know that system 2.7 is compatible.Therefore, system 2.7 could be used to obtain through a line integral a first integral of 2.4 associated with μ.
Similarly, for second-order ODEs, there is a correspondence between integrating factors and Lie point symmetries in the sense of Theorem 7 in 11 .
, and any solution of the first-order linear system is an integrating factor of 2.4 .Now, we briefly summarize the approach which is presented in 10 .
The linearized ODE for 2.4 is given by and the corresponding adjoint ODE is The solutions μ t, x, ẋ of ODE 2.10 , holding for any x t satisfying the second-order ODE 2.4 , are the adjoint symmetries of 2.4 .The adjoint invariance conditions for μ t, x, ẋ to be an integrating factor of 2.4 are Equations 2.11 -2.12 must hold for arbitrary values of t, x, ẋ.

Li énard Type Oscillator Equation
Liénard type nonlinear oscillators of the form ẍ kx ẋ k 2 9 x 3 ax 0 3.1 and their generalizations are widely used in applications in the context of nonlinear oscillations 14-18 .
We know that 3.1 , that is, admits the vector field v ∂/∂x as a λ-symmetry, where λ is any particular solution to the following equation: Therefore, we obtain the following equation: x 2 a 0, 3.4 that corresponds to 3.3 .
For the sake of simplicity, we try to find a solution λ of 3.4 of the form λ t, x, ẋ λ 1 t, x ẋ λ 2 t, x .This ansatz leads to the following system: x 2 a 0.

3.5
A particular solution of the second equation is given by λ 1 1/x.The first and third equations become

3.6
The general solution of the first equation is given by λ 2 t, x −kx/3 x −2 λ 21 t .Since the last equation becomes x −1 λ 21 t x −3 λ 21 t 2  1/3 kλ 21 t 0, we can choose λ 21 t 0. In consequence, v ∂/∂x is a λ-symmetry for λ t, x, ẋ 1/x ẋ − kx/3.In order to find an integrating factor associated to λ, we must find a first-order invariant w t, x, ẋ of v λ, 1 .The equation that corresponds to 2.5 is It is clear that w t, x, ẋ − 1/x ẋ − kx/3 is a solution of 3.7 .In terms of {t, w, w t }, 3.1 becomes w t − w 2 − a 0, the general solution of which is given by 1/ √ a Arctan w/ √ a t c, c ∈ Ê.According to 2.6 , an integrating factor is given by

3.8
We observe that the method we have followed not only provides the integrating factor but also gives the conserved form of the equation without additional computation.The conserved form of the resulting equation is given by 3.9 We note that 3.1 is a particular case of a family of equations ẍ a 1 x ẋ a 0 x 0, 3.10 with a 1 x kx, a 0 x k 2 /9 x 3 ax appearing in 19 .Muriel and Romero in that reference suggest a well-defined algorithm through general Sundman transformations for the class of nonlinear second-order ODEs.They propose considering the problem of linearization through nonlocal transformations from the point of the λ-symmetries admitted by the equation and their associated first integrals.In our study without considering nonlocal transformations, integrating factors and the associated first integrals are derived from the method to compute λ-symmetries and the associated reduction algorithm.Moreover, our results 3.8 and 3.9 are compatible with the results obtained in 19 .Now, we try to obtain the integrating factor of 3.1 with the method of 10 .The ODE 3.1 is not self-adjoint, so that its adjoint symmetries are not symmetries.Here, the adjoint symmetry determining equation 2.11 for μ t, x, ẋ is the following;

3.11
The extra adjoint-invariance determining equation 2.12 becomes If we consider the special case μ μ t, x , then 3.11 becomes

3.13
Seperating by monomials, that is, ẋ2 : we get μ 0. The situations for μ μ x, ẋ and μ μ t, ẋ are a more difficult task than solving the original ODE.

Frequency-Damped Oscillator Equation
We consider the frequency-damped oscillator equation ẍ x ẋ2 0 3.15 studied by Gordon 20 ,Sarlet et

3.23
We note that 3.15 is a particular case of a family of equations ẍ a 2 x ẋ2 a 1 t ẋ 0, 3.24 with a 2 x x, a 1 t 0 appearing in 23 .The authors characterized the 3.24 that admits first integrals of the form A t, x ẋ B t, x through an easy-to-check criterion expressed in terms of functions S 1 , S 2 , S 3 , and S 4 given by 3.3 and 3.4 -3.6 in 23 .We could obtained our results 3.21 and 3.23 using this novel approach.Moreover, 3.15 was studied by the same authors from the point of view of linearizations through local and nonlocal transformations in 24 .Our results 3.21 and 3.23 are compatible with the results obtained in 24 .

Conclusions
In this paper, we derived integrating factors and first integrals for Liénard type and frequency-damped oscillator equations by λ-symmetry approach.From the λ-symmetry and the associated algorithm of reduction, the integrating factor 3.8 and the associated first integral 3.9 were derived for Liénard type equation.We also show the difficulties to get integrating factors for Liénard type equation by the other methods.
The knowledge of a λ-symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems 2.8 of partial differential equations.Solving this system, we obtain the integrating factor 3.21 and the associated first integral 3.23 for frequency-damped oscillator equation 3.15 .