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

Abstract

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.

1. 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]. Several authors have obtained necessary and sufficient conditions for a function 𝜇(𝑡,𝑥,𝑥̇) to be an integrating factor of a second-order ODEs. Most of their approaches rest on the fact that the multiplication of function 𝜇 with second-order ODE is a total derivative, and therefore its variational derivative is null. As a consequence, integrating factors can be determined as solutions of a second-order linear system of partial differential equations (PDEs). Since solving this system is usually a more difficult task than solving the original ODE, many studies have been done to investigate special classes of integrating factors, through specific ansatze for 𝜇 [9]. For instance, in [10] Anco and Bluman derive integrating factors as the solutions of the adjoint equation of the linearized equation and an additional equation that describes an extra adjoint-invariance condition.

In their recent papers, Muriel and Romero [11, 12] presented what they termed a systematic algorithm for the construction of integrating factors of the form 𝜇(𝑡,𝑥,𝑥̇)for second-order ODEs. With this new powerful method, integrating factors and the associated first integrals are derived from the method to compute 𝜆-symmetries and the associated 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.

2. 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 equationΔ𝑡,𝑥^(𝑛)=0,(2.1) where variables (𝑡,𝑥) are in some open set 𝑀⊂𝑇×𝑋≃ℝ². For 𝑘∈ℕ, 𝑀^𝑘⊂𝑇×𝑋^(𝑘) denotes the corresponding 𝑘-jet space, and the elements of 𝑀^(𝑘) are denoted by (𝑡,𝑥^(𝑘))=(𝑡,𝑥,𝑥̇,…,𝑥^𝑘).

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)),𝜇𝑡,𝑥^(𝑘)⋅Δ𝑡,𝑥^(𝑛)=𝐷_𝑡Δ𝑡,𝑥^(𝑛−1).(2.2)

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∈ℝ, appears as a consequence of the reduction algorithm associated to that 𝜆-symmetry [13]. If 𝜆∈𝐶^∞(𝑀^(𝑘)), 0≤𝑘≤𝑛−1,is any solution of the partial differential equation^𝑛−1_𝑖=0𝐷_𝑡+𝜆^𝑖(1)𝜕Δ𝜕𝑥^(𝑖)=0when𝑥^(𝑛)=𝐹𝑡,𝑥^(𝑛−1),(2.3) then the vector field 𝑣=𝜕_𝑥is 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)𝜕_̇𝑥 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 equation𝑤_𝑥+𝑤_̇𝑥𝜆(𝑡,𝑥,𝑥̇)=0.(2.5) 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)=𝐶, 𝐶∈ℝ. It is clear that 𝐷_𝑡(𝐺(𝑡,𝑤(𝑡,𝑥,𝑥̇))=0 is an equivalent form of (2.4). Therefore,𝜇(𝑡,𝑥,𝑥̇)=𝐺_𝑤(𝑡,𝑥,𝑤(𝑡,𝑥,𝑥̇))⋅𝑤_̇𝑥(𝑡,𝑥,𝑥̇)(2.6) 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].

Theorem 2.2 (see [11]). (a) If 𝐼(𝑡,𝑥,𝑥̇) is a first integral of (2.4), then the vector field 𝑣=𝜕_𝑥 is a 𝜆-symmetry of (2.4) for 𝜆=−𝐼_𝑥/𝐼_̇𝑥 and 𝑣^[𝜆,(1)]𝐼=0.
(b) Conversely, if 𝑣=𝜕_𝑥 is a 𝜆-symmetry of (2.4) 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_̇𝑥 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_̇𝑥=−𝜆𝜇 and the system𝐼_𝑥=𝜇(𝜆𝑥̇−𝜙),𝐼_𝑥=−𝜆𝜇,𝐼_̇𝑥=𝜇(2.7) 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].

Theorem 2.3 (see [11]). If 𝑣 is a Lie point symmetry of (2.4) and 𝑄=𝜂−𝜉𝑥̇ is its characteristic, then 𝑣=𝜕_𝑥 is a 𝜆-symmetry of (2.4) for 𝜆=𝐴(𝑄)/𝑄, and any solution of the first-order linear system 𝐴(𝜇)+𝐹_̇𝑥−𝐴(𝑄)𝑄𝜇=0,𝜇^𝑥+𝐴(𝑄)𝑄𝜇_̇𝑥=0(2.8) 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 𝐿[𝑥]𝜎=𝑑²𝜎𝑑𝑡²−𝐹_̇𝑥𝑑𝜎𝑑𝑡−𝐹^𝑥𝜎=0,(2.9) and the corresponding adjoint ODE is 𝐿^∗[𝑥]𝜛=𝑑²𝜛𝑑𝑡²+𝑑𝑑𝑡𝐹^̇𝑥𝜛−𝐹_𝑥𝜛=0.(2.10) The solutions 𝜛=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 𝜇_𝑡𝑡+2𝑥̇𝜇_𝑡𝑥+2𝐹𝜇_𝑡̇𝑥+(𝑥̇)²𝜇_𝑥𝑥+2𝑥̇𝐹𝜇_𝑥̇𝑥+𝐹²𝜇_̇𝑥̇𝑥+𝐹_𝑡+𝑥̇𝐹_𝑥+2𝐹𝐹_̇𝑥𝜇_̇𝑥𝜇+𝐹+𝑥̇𝐹_̇𝑥𝜇_𝑥+𝐹_̇𝑥𝜇_𝑡+𝐹_𝑡̇𝑥+𝑥̇𝐹_̇𝑥̇𝑥+𝐹𝐹_̇𝑥̇𝑥−𝐹_𝑥𝜇=0,(2.11)_𝑡̇𝑥+𝑥̇𝜇_𝑥̇𝑥+𝐹𝜇_̇𝑥̇𝑥+2𝐹_̇𝑥𝜇_̇𝑥+2𝜇_𝑥+𝐹_̇𝑥̇𝑥𝜇=0.(2.12) Equations (2.11)-(2.12) must hold for arbitrary values of 𝑡,𝑥,𝑥̇.

3. 𝜆-Symmetries, Integrating Factors, and First Integrals

3.1. Liénard Type Oscillator Equation

Liénard type nonlinear oscillators of the form𝑥̈+𝑘𝑥𝑥̇+𝑘²9𝑥³+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,𝑥̈=−𝑘𝑥𝑥̇−𝑘²9𝑥³−𝑎𝑥=𝐹,(3.2) admits the vector field 𝑣=𝜕/𝜕𝑥as a 𝜆-symmetry, where 𝜆 is any particular solution to the following equation:𝐷_𝑡+𝜆²(1)=¹_𝑖=0𝐷_𝑡+𝜆^𝑖(1)𝜕𝐹𝜕𝑥^(𝑖)when𝑥̈=−𝑘𝑥𝑥̇−𝑘²9𝑥³−𝑎𝑥=𝐹,(3.3) or equivalently, 𝐷_𝑡𝜆+𝜆²=𝜆_𝑡+𝑥̇𝜆_𝑥+(−𝑘𝑥𝑥̇−(𝑘²/9)𝑥³−𝑎𝑥)𝜆_̇𝑥+𝜆²=−𝑘𝑥̇−(𝑘²/3)𝑥²−𝑎−𝑘𝜆𝑥. Therefore, we obtain the following equation:𝜆_𝑡+𝑥̇𝜆_𝑥+−𝑘𝑥𝑥̇−𝑘²9𝑥³−𝑎𝑥𝜆_̇𝑥+𝜆²+𝑘𝜆𝑥+𝑘𝑥̇+𝑘²3𝑥²+𝑎=0,(3.4) that corresponds to (3.3).

For the sake of simplicity, we try to find a solution 𝜆of (3.4) of the form 𝜆(𝑡,𝑥,𝑥̇)=𝜆₁(𝑡,𝑥)𝑥̇+𝜆₂(𝑡,𝑥). This ansatz leads to the following system:𝜆_1𝑡+𝜆𝜆_2𝑥+2𝜆₁𝜆₂+𝑘=0,_1𝑥𝜆+𝜆²¹=0,_2𝑡−𝑘²9𝜆¹𝑥³−𝑎𝜆₁𝑥+𝜆²²+𝑘𝜆₂𝑥+𝑘²3𝑥²+𝑎=0.(3.5) A particular solution of the second equation is given by 𝜆₁=1/𝑥. The first and third equations become𝜆_2𝑥+2𝑥𝜆²𝜆+𝑘=0,_2𝑡+2𝑘²9𝑥²+𝜆²²+𝑘𝜆₂𝑥=0.(3.6) The general solution of the first equation is given by 𝜆₂(𝑡,𝑥)=−𝑘𝑥/3+𝑥⁻²𝜆₂₁(𝑡). Since the last equation becomes 𝑥⁻¹𝜆^21(𝑡)+𝑥⁻³𝜆₂₁(𝑡)²+(1/3)𝑘𝜆₂₁(𝑡)=0, we can choose 𝜆₂₁(𝑡)=0.In consequence, 𝑣=𝜕/𝜕𝑥is 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 𝑤_𝑥+1𝑥𝑥̇−𝑘3𝑥𝑤^̇𝑥=0.(3.7) 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−𝑤²−𝑎=0, the general solution of which is given by (1/√𝑎)Arctan(𝑤/√𝑎)=𝑡+𝑐, 𝑐∈ℝ.According to (2.6), an integrating factor is given by𝜇(𝑡,𝑥,𝑥̇)=𝐺_𝑤(𝑡,𝑥,𝑤(𝑡,𝑥,𝑥̇))⋅𝑤_̇𝑥=−1𝑎+(𝑥̇/𝑥+𝑘𝑥/3)²𝑥.(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 𝐷_𝑡√1𝑎Arctan−√1𝑎𝑥𝑥̇−3𝑘√𝑥𝑎=0.(3.9)

We note that (3.1) is a particular case of a family of equations 𝑥̈+𝑎₁(𝑥)𝑥̇+𝑎₀(𝑥)=0,(3.10) with 𝑎₁(𝑥)=𝑘𝑥, 𝑎₀(𝑥)=(𝑘²/9)𝑥³+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 𝜛=𝜇(𝑡,𝑥,𝑥̇) is the following;𝜇_𝑡𝑡+2𝑥̇𝜇_𝑡𝑥+2−𝑘𝑥𝑥̇−𝑘²9𝑥³−𝑎𝑥𝜇_𝑡̇𝑥+𝑥̇²𝜇_𝑥𝑥+2𝑥̇−𝑘𝑥𝑥̇−𝑘²9𝑥³−𝑎𝑥𝜇_𝑥̇𝑥+−𝑘𝑥𝑥̇−𝑘²9𝑥³−𝑎𝑥²𝜇_̇𝑥̇𝑥+𝑥̇−𝑘𝑥̇−13𝑘²𝑥²−𝑎+2−𝑘𝑥𝑥̇−𝑘²9𝑥³−𝑎𝑥(−𝑘𝑥)𝜇_̇𝑥+−𝑘𝑥𝑥̇−𝑘²9𝑥³−𝑎𝑥+𝑥̇(−𝑘𝑥)𝜇_𝑥+(−𝑘𝑥)𝜇_𝑡+−𝑘𝑥̇−−𝑘𝑥̇−13𝑘²𝑥²−𝑎𝜇=0.(3.11) The extra adjoint-invariance determining equation (2.12) becomes𝜇_𝑡̇𝑥+𝑥̇𝜇_𝑥̇𝑥+−𝑘𝑥𝑥̇−𝑘²9𝑥³−𝑎𝑥𝜇_̇𝑥̇𝑥+2(−𝑘𝑥)𝜇_̇𝑥+2𝜇_𝑥=0.(3.12) If we consider the special case 𝜇=𝜇(𝑡,𝑥), then (3.11) becomes𝑥̇²𝜇_𝑥𝑥+𝑥̇2𝜇_𝑡𝑥−𝑘𝑥𝜇_𝑥−𝑘𝑥𝜇_𝑥+𝜇_𝑡𝑡−𝑘²9𝑥³+𝑎𝑥𝜇_𝑥−𝑘𝑥𝜇_𝑡+13𝑘²𝑥²+𝑎𝜇=0.(3.13) Seperating by monomials, that is,𝑥̇∶2𝑥𝜇̇²_𝑡𝑥∶𝜇_𝑥𝑥=0,1∶𝜇_𝑡𝑡−2𝑥𝑘𝜇_𝑥=0,−𝑘²9𝑥³+𝑎𝑥𝜇_𝑥−𝑘𝑥𝜇_𝑡+13𝑘²𝑥²+𝑎𝜇=0,(3.14) 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.

3.2. Frequency-Damped Oscillator Equation

We consider the frequency-damped oscillator equation𝑥̈+𝑥𝑥̇²=0(3.15) studied by Gordon [20], Sarlet et al. [21], and Mimura and Nono [22].

It is clear that 𝑣=𝜕/𝜕𝑡 is a Lie point symmetry of (3.15). The vector field associated with (3.15) is 𝐴=𝜕𝜕𝑡+𝑥̇𝜕𝜕𝑥−𝑥𝑥̇²𝜕𝜕𝑥̇,(3.16) and the characteristic of 𝑣 is 𝑄=−𝑥̇. The corresponding second equation of (2.8) becomes𝜇_𝑥−𝑥𝑥̇𝜇_̇𝑥−𝑥𝜇=0.(3.17) The solution of (3.17) can be obtained by the characteristic method of Lagrange𝜇(𝑡,𝑥,𝑥̇)=𝑒^𝑥2/2𝑀𝑡,𝑥̇𝑒^𝑥2/2,(3.18) where 𝑀 is an arbitrary function of 𝑡 and 𝑤=𝑥̇𝑒^𝑥2/2.Since 𝜇 must also satisfy the first equation in (2.8),𝑀_𝑡(𝑡,𝑤)=0.(3.19) This implies that 𝑀 depends only on 𝑤 and 𝜇(𝑡,𝑥,𝑥̇)=𝑒^𝑥2/2𝑀𝑥̇𝑒^𝑥2/2(3.20) is an integrating factor of (3.15), where 𝑀 is an arbitrary function.

In order to find a first integral 𝐼 of (3.15) such that 𝐼_̇𝑥=𝜇, we must solve the system that corresponds to (2.7). If, for example, we choose 𝑀(𝑤)=𝑤, then we get the particular integrating factor𝜇(𝑡,𝑥,𝑥̇)=𝑥̇𝑒^𝑥2.(3.21) The above integrating factor is different from the previously derived integrating factors for (3.15) by Anco and Bluman [10].

Integrating factor (3.21) of (3.15), the corresponding system, (2.7), becomes 𝐼_𝑡=0,𝐼_𝑥=𝑥𝑥̇²𝑒^𝑥2,𝐼_̇𝑥=𝑥̇𝑒^𝑥2.(3.22) By evaluating the corresponding line integral, we get the general solution of (3.22) and a class of first integrals of (3.15), 𝐼(𝑡,𝑥,𝑥̇)=𝑥̇²2𝑒^𝑥2+𝐶,(𝐶∈ℝ).(3.23) We note that (3.15) is a particular case of a family of equations 𝑥̈+𝑎₂(𝑥)𝑥̇²+𝑎₁(𝑡)𝑥̇=0,(3.24) with 𝑎₂(𝑥)=𝑥, 𝑎₁(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 𝑆₁, 𝑆₂, 𝑆₃, and 𝑆₄ 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].

4. 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).

Acknowledgment

The author would like to thank the reviewers for their comments that helped improve the manuscript.