## Abstract

We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ODEs. We establish an important interconnection between the extended Prelle–Singer procedure and λ-symmetries approach applicable to third-order ODEs to bring out the various linkages associated with these different techniques. By establishing this interconnection we demonstrate that given any one of the quantities as a starting point in the family consisting of Jacobi last multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, λ-symmetries, integrating factors and null forms one can derive the rest of the quantities in this family in a straightforward and unambiguous manner. We also illustrate our findings with three specific examples.

## 1. Introduction

In our previous paper [1], we established a connection between the extended Prelle–Singer procedure and five other analytical methods which are widely used to identify integrable systems described by second-order ODEs and demonstrated the interconnections between Lie point symmetries, λ-symmetries, Darboux polynomials, Prelle–Singer procedure, adjoint-symmetries and the Jacobi last multiplier (JLM). We also illustrated the interconnections by considering a nonlinear oscillator equation as an example. A natural question which arose was whether these interconnections also exist in higher order ODEs. In this paper, we consider this problem for third-order ODEs and find certain new and interesting results.

We start our investigation with the following question. Given any one of the quantities as a starting point in the family consisting of multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, λ-symmetries, integrating factors and null forms can one obtain the rest of all the quantities in this family? In the case of third-order ODEs exploring the interconnections between the above-mentioned quantities is difficult both conceptually as well as technically, which will become clear as we proceed. In fact, we get an affirmative answer to our questions. We divide our analysis into two parts. In the first part, as we did in the case of second-order ODEs, we try to interlink the quantities by introducing suitable transformations in the extended Prelle–Singer procedure. Once the interconnections are identified, in the second part, we demonstrate that starting from any one of the quantities in this family one can derive all the other quantities in a simple and unambiguous way.

In the first part, the interplay between various methods is demonstrated by establishing a road map between the extended Prelle–Singer procedure and the other methods, namely (i) the JLM, (ii) Darboux polynomials, (iii) Lie point symmetries, (iv) adjoint-symmetries, and (v) λ-symmetries. In the above, we have listed the methods in chronological order. In other words, the JLM was invented first and the last one to be invented is the extended Prelle–Singer procedure. So, naturally, we try to accommodate other methods into the extended Prelle–Singer procedure. In the later procedure, one has three equations to integrate to obtain the null forms *U* and *S* and the integrating factor *R*. To interconnect these quantities, we introduce three new variables, *V* , *X* and *F*, which are to be determined through the transformations *U*=−*D*[*V* ]/*V* , *S*=*X*/*V* and *R*=*V*/*F*, so that one can rewrite the determining equations for *U*, *S* and *R* in terms of *X*, *V* and *F*, respectively. Doing so, we find that the determining equation for the function *F* exactly coincides with the determining equation for the Darboux polynomials, which in turn establishes a connection between the integrating factors and the Darboux polynomials. Now properly combining the expressions which arise in the extended Prelle–Singer procedure with the λ-symmetry determining the equation we obtain the following relation between *U* and *S* with λ (see §3 for details): *U*=−(*D*[λ]+λ^{2}+*S*)/λ. This interlink is vital and new to the literature. Since the null forms *U* and *S* are now expressed in terms of *V* and *X*, this interlink can also be written in terms of *V* and λ. By doing so, we obtain an equation solely in terms of *V* and λ, that is, *D*^{2}[*V* ]−*D*[*V* ](*V* (*D*[λ]+λ^{2}−*V* is known or vice versa (here *ϕ*(*t*,*x*,*V* can be determined by solving this equation and from the latter *X* can be fixed, and so *U*, *S* and *R* can be determined. The Darboux polynomials can be determined either from the multipliers or from their own determining equations. This in turn establishes the interconnection between all these methods. One important result which we observe in this part is that, for a given λ, the null forms *U* and *S* are not unique, which can be seen from the above relation. Because of this, one may find multiple null forms for a given equation and one has to choose the correct form of *U* and *S* in order to obtain an independent integral. From the remaining forms of *U* and *S* one may obtain other useful information such as symmetries, integrating factors, Darboux polynomials and the JLM; however, the resulting integrals turn out to be dependent ones. This feature is different from second-order ODEs, where one has one-to-one correspondence between the null forms and λ-symmetries (λ=−*S*). We illustrate this important point by considering three examples.

Next, we move on to how, given any one of the system quantities, the remaining quantities in this family can be derived. Using the interconnections which we identified above, and starting from any one of the quantities, we establish road maps to connect to all the other quantities. After a detailed analysis, we find that it is sufficient to consider any one of the following three cases as starters: (i) Lie point symmetries, (ii) integrating factors or (iii) Darboux polynomials. Then the remaining quantities can be uniquely determined. The other three possibilities, namely (iv) λ-symmetries, (v) null forms, and (vi) the JLM, are only subcases of the previous three cases, respectively. The exact road map in all the cases is explicitly demonstrated in §4. To illustrate the ideas introduced in this paper we consider three examples.

Since some of the methods applicable for the third-order ODEs are not much discussed in the literature, as in the case of second-order ODEs, to begin with we review briefly the six different analytical methods in the next section (§2) in order to be self-contained.

## 2. Material and methods

In this section, we briefly recall the six analytical methods with reference to the third-order ODEs which are widely used in the contemporary literature to derive one or more of the following quantities: (i) multipliers, (ii) integrating factors, (iii) symmetries, and (iv) integrals. We present the methods in chronological order.

### (a) Jacobi last multiplier method

The JLM method was introduced by Jacobi in 1844 [2,3], but remained dormant for a long time. Nucci & Leach [4] have demonstrated the applicability of this method in exploring non-standard Lagrangians associated with certain second-order nonlinear ODEs. The attractive feature of the method is that if we know two independent JLMs *M*_{1} and *M*_{2} of the given equation, then their ratio yields a first integral for the given equation.

We consider a third-order ODE,
*ϕ* is a function of *t*,*x*,*t*-dependence:
*f*, *g* and *h* are suitably chosen functions of *x*, *y* and *z*, respectively. The analysis can be extended in principle to the general case with *t*-dependence included, though it becomes more involved now (e.g. [5]). We consider two integrals *I*_{1}(*x*,*y*,*z*) and *I*_{2}(*x*,*y*,*z*) of (2.2) whose total differentials are given by
*f*_{1}(*x*,*y*,*z*), *f*_{2}(*x*,*y*,*z*) and *f*_{3}(*x*,*y*,*z*). We assume that these equations are those coming out from the original equation (2.2) after multiplying by an integrating factor *M*(*x*,*y*,*z*). Then comparing equations (2.2) and (2.4), we have
*M*:
*f*(*x*,*y*,*z*)=*y*, *g*(*x*,*y*,*z*)=*z* and *h*(*x*,*y*,*z*)=*ϕ*, equation (2.8) can be written as
*ϕ* in (2.1) also depends on *t* explicitly, the above operator *D*=∂/∂*t*+*x*)+*y*)+ż(∂/∂*z*).

An important application of multipliers is that one can determine the integrals associated with the given equation by evaluating their ratios. If *M*_{1} and *M*_{2} are two multipliers, then it is straightforward to check from (2.9) that one can identify an integral as *I*=*M*_{1}/*M*_{2} [6]. So if we have a sufficient number of multipliers, we can obtain the necessary integrals to prove the integrability of (2.1).

### (b) Darboux polynomials approach

The Darboux polynomials method was developed by Darboux in 1878 [7]. It provides a strategy to find first integrals. Darboux showed that if we have *n*(*n*+1)/2+2 Darboux polynomials, where *n* is the order of the given equation, then there exists a rational first integral which can be expressed in terms of these polynomials. In the case of third-order ODEs, we can get eight Darboux polynomials for a given equation [7].

The Darboux polynomials determining equation is given by the following expression:
*D* is the total differential operator and *g*(*t*,*x*,*F*, and an integral *I* of (2.1), the quantity *f*(*I*)*F*, where *f* is arbitrary, is also a solution of (2.10) for the same cofactor. Using this fact and solving equation (2.10), we can obtain the Darboux polynomials (*F*) and the cofactors (*g*); see Mohanasubha *et al.* [8] for further details on the method.

### (c) Lie symmetry analysis

Lie symmetry analysis is one of the powerful methods to investigate the integrability property of the given ODE of any order *n* where one first explores the symmetry vector fields associated with it. The Lie symmetry vector fields can then be used to derive integrating factors, conserved quantities and so on [9].

We consider a third-order ODE of the form (2.1). The invariance of equation (2.1) under a one-parameter group of Lie point symmetries, corresponding to infinitesimal transformations,
*ξ*(*t*,*x*) and *η*(*t*,*x*) are the coefficient functions of the associated generator of an infinitesimal transformation and *ε* is a small parameter, is specified by
*η*^{(1)}, *η*^{(2)} and *η*^{(3)} are the first, second and third prolongations, respectively, of the infinitesimal point transformations (2.11) and are defined to be
*t*. Substituting the known expression *ϕ* in (2.12) and solving the resultant equation, we can get the Lie point symmetries associated with the given third-order ODE. The maximum number of admissible Lie point symmetries for a third-order ODE (2.1) is seven [9]. The associated vector field is given by *Ω*=*ξ*(∂/∂*t*)+*η*(∂/∂*x*).

One may also introduce a characteristic *Q*=*η*−*ξ* and rewrite the invariance condition (2.12) in terms of a single variable *Q* in the form
*ξ* and *η* associated with the Lie point symmetries from the class of solutions to (2.14), which depends only on *x* and *t* and also has a linear dependence in

### (d) Adjoint-symmetries

In general, for systems of one or more ODEs, an integrating factor is a set of functions, multiplying each of the ODEs, which yields a first integral. If the system is self-adjoint (i.e. the adjoint of the linearized symmetry condition is the same as the linearized symmetry condition) then its integrating factors are necessarily solutions of its linearized system. Such solutions are also the symmetries of the given system of ODEs. If a given system of ODEs is not self-adjoint, then its integrating factors are necessarily solutions of the adjoint system of its linearized system. These solutions are known as adjoint-symmetries of the given system of ODEs.

The adjoint ODE of the linearized symmetry condition (2.14) can be written as [10]
*D* is the total derivative operator which is given by *D*=∂/∂*t*+*x*)+*ϕ*(∂/∂*Λ*(*t*,*x*,

### (e) λ-symmetries

All the nonlinear ODEs do not necessarily admit Lie point symmetries. Under such a circumstance, one may look for generalized symmetries (other than Lie point symmetries) associated with the given equation. One such generalized symmetry is the λ-symmetry. Let *t*,*x*,*ξ*(∂/∂*t*)+*η*(∂/∂*x*)+*η*^{[λ,(1)]}(∂/∂*η*^{[λ,(2)]}(∂/∂*η*^{[λ,(3)]}(∂/∂*η*^{[λ,(1)]}, *η*^{[λ,(2)]} and *η*^{[λ,(3)]} are first, second and third λ-prolongations, respectively, whose explicit expressions are given by [11–13]
*ξ*,*η* and λ. If the given ODE admits Lie point symmetries, then the λ-symmetries can be derived without solving the λ-prolongation condition. In this case, the λ-symmetries can be deduced from the relation
*D* is the total differential operator and *Q*=*η*−*ξ*

### (f) Extended Prelle–Singer method

In a series of papers Chandrasekar, Senthilvelan and Lakshmanan have developed a stand-alone method, namely the extended Prelle–Singer procedure, to investigate the integrability of the given ODE which may be of any order, including coupled ones. In the following, we recall briefly the extended Prelle–Singer procedure which is applicable for third-order ODEs [14].

We assume that the ODE (2.1) admits a first integral *I*(*t*,*x*,*C*, with *C* constant on the solutions, so that the total differential gives
*ϕ*) *dt*−*d**U*(*t*,*x*,*dt*−*U*(*t*,*x*,*d**S*(*t*,*x*,*dt*−*S*(*t*,*x*,*dx* to the latter, we obtain that on the solutions the 1-form
*R*(*t*,*x*,*I*), integrating factor(*R*) and the null term(*S*):
*S*, *U* and *R*, we impose the compatibility conditions *I*_{tx}=*I*_{xt}, *D*=∂/∂*t*+*x*)+*ϕ*(∂/∂*S*, *U* and *R*. From the known expressions, *S*,*U* and *R*, we can determine the integrals which appear on the left-hand side of equation (2.24) by straightforward integration.

Integrating equation (2.24), we find
*S*,*U*,*R*) in (2.31) defines a first integral.

## 3. Interconnections

In the previous section, we have discussed six specific analytical methods which are used to derive integrating factors, symmetries of various kinds, null forms and integrals associated with the third-order ODEs. A question which we raise now is what is the interconnection between these various methods, that is, given any one of the quantities in the family, say multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, λ-symmetries or integrating factors and null forms, can one obtain the rest of the quantities in this family? To answer this question, we can explore the hidden interconnections that exist between these functions and interlink all these methods. To achieve this task we introduce certain transformations in the extended Prelle–Singer procedure which in turn connect globally all the above-mentioned quantities. The details are given below (it may be noted that one can establish the same interconnection by taking any one of the other methods as the starting point).

With the aid of the following transformations on the null forms of the Prelle–Singer method, see equation (2.25):
*V* (*t*,*x*,*X*(*t*,*x*,*D* is the total differential operator, equations (2.26) and (2.25), respectively, can be rewritten in the form
*F*(*t*,*x*,*F* as
*g*=

We mention here that, once we know the functions *U* and *S*, the integrating factor *R* can be determined within the Prelle–Singer procedure itself. But to connect the integrating factors to Darboux polynomials these transformations are essential. More importantly the connection between the null forms and λ-symmetries can be unearthed through the function *V* which appears in the transformations (3.1) and (3.4), as we see below.

### (a) Connection between λ-symmetries and null forms

Now we investigate how these expressions, namely *U* and *S*, are interconnected with λ-symmetries. In the case of second-order ODEs, the λ-symmetry is nothing but the null form with a negative sign [12]. This one-to-one correspondence came from the result that the *S*-determining equation in the Prelle–Singer procedure differs only by a negative sign from that of the λ-symmetry determining equation [1]. However, in the case of third-order ODEs, we have two null forms *S* and *U* which have to be connected to a single function λ as demonstrated below.

Let *I*(*t*,*x*,*R*=−*I*/d*t*=0 gives
*I*(*t*,*x*,*t*,*x*,*ξ*(∂/∂*t*)+*η*(∂/∂*x*)+*η*^{[λ,(1)]}(∂/∂*η*^{[λ,(2)]}(∂/∂*η*^{[λ,(1)]} and *η*^{[λ,(2)]} are the first and second λ-prolongations, respectively, whose explicit expressions are given in the first two expressions of (2.18). Expanding equation (3.7), we get the following expression for λ corresponding to *η*^{(1)} and *η*^{(2)} are the first and second Lie point prolongations, respectively, which are given in equation (2.13). For the vector field

Now we connect this expression which comes out from the λ-symmetry analysis with the null forms in the extended Prelle–Singer procedure. In this regard, we can deduce the following expressions from the last three equations of (2.24):
*U* and *S* with λ in the form
*U* and *S*, are connected with the λ-symmetries through a differential relation. This interconnection is demonstrated for the first time in the literature. We mention here that, while deriving the relation (3.11), we assumed that λ≠0. When λ=0, we have *I*_{x}=0 (see equation (3.9)) and in this case one of the null forms (*S*) vanishes. This result is also consistent with the extended Prelle–Singer procedure.

For practical purposes, we can rewrite the relation (3.11) in terms of a single variable, say in *V*, as follows. Using (3.2), we can express *X* in terms of *V* and, substituting the latter in the second expression in (3.1), we find *S* in terms of *V*. The resultant expression reads
*S* and *U*, which appear in (3.11), by *V*, we find
*V*, which in turn unambiguously fixes the null forms *U* and *S* through the relations given in equation (3.1). In other words, one can get the null forms *U* and *S* from λ by finding the function *V* also. In this sense, equation (3.13) may also be treated as a ‘second bridge’ which connects λ-symmetries with null forms. If we already know the null forms *U* and *S*, equation (3.11) yields the λ-symmetries in a straightforward manner. In this sense, one can determine (i) λ from *U* and *S* and (ii) *U* and *S* from λ. The expressions (3.11) and (3.13) interconnect the Prelle–Singer procedure and the λ-symmetry analysis.

### (b) Connection between Lie point symmetries and null forms

The relation between Lie point and λ-symmetries has already been established (see equation (2.20)). Substituting this in (3.11), we can obtain an expression that relates Lie point symmetries with the null forms in the following manner:
*Q*=*η*−*ξ* is the characteristics.

For the sake of completeness, in the following, we recall the interconnection that is already known in the literature.

### (c) Connection between the Jacobi last multiplier and Darboux polynomials/ integrating factors

Using expression (3.4), we can relate the Darboux polynomials with the integrating factor and in fact the denominator of (3.4) is nothing but the Darboux polynomials.

By comparing equations (2.10) and (2.9), we can relate the Darboux polynomials with the JLM as
*R* in the Prelle–Singer method is the product of the function *V* and the JLM *M*, that is, *R*=*V* *M*.

### (d) Connection between adjoint-symmetries and integrating factors

We rewrite equations (2.25)–(2.27) as a single equation in one variable, for example in *R*. Then the resultant equation reads
*R* is nothing but the adjoint-symmetry *Λ*, that is,

### (e) Connection between Lie point symmetries and Jacobi last multipliers

The connection between Lie point symmetries and JLM has been known for a long time in the form *M*=1/Δ, where
*ξ*_{1},*η*_{1}), (*ξ*_{2},*η*_{2}) and (*ξ*_{3},*η*_{3}) are three sets of Lie point symmetries (see below) of the third-order ODE, *i*=1,2, are their corresponding first and second prolongations, respectively, and the inverse of Δ becomes the multiplier of the given equation [6].

### (f) Comparison between interconnections for second- and third-order ODEs

In the above-said interconnections, some of the interconnections are common to both second- and third-order nonlinear ODEs except for their orders, while the others are different. Such common and differing connections are listed below.

(1)

*Common features*. The common connections are the ones between (i) Lie point symmetries and λ-symmetries, (ii) Lie point symmetries and the JLM, (iii) adjoint-symmetries and integrating factors, and (iv) Darboux polynomials and the JLM.(2)

*Differing features*.(i) The uncommon relation is the connection between λ-symmetries and null forms. In the cases of second- and third-order ODEs, the connections between λ-symmetries and null forms are entirely different. In the case of second-order ODEs, the λ-symmetry is nothing but the null form with a negative sign. This one-to-one correspondence came from the result that the

*S*-determining equation in the Prelle–Singer procedure differs only by a negative sign from that of the λ-symmetry determining equation [12]. In other words, there is a one-to-one correspondence between the λ-symmetries and the null form*S*. However, in the case of third-order ODEs, we have two null forms (*S*and*U*) which have to be connected to the single function λ. Our analysis shows that these two null forms are connected with the λ-symmetry through a single expression.(ii) The connection between characteristics and null forms is also different in the case of second- and third-order ODEs. In the case of second-order ODEs, there exists only one null form which is directly connected with the characteristics, while in the case of third-order ODEs the equation which connects the characteristics and null forms contains both the null forms.

## 4. Illustration

### (a) Example 1

We consider the following third-order nonlinear ODE:
*et al.* [17] and Ibragimov & Meleshko [18] in the form *c*(^{2}/*x*=0 has been considered by Euler & Euler [19], who showed that it can be linearized to a free particle equation through a Sundman transformation. Equation (4.1) was studied by Chandrasekar *et al.* [20] from the integrability point of view using the Prelle–Singer procedure. Here we consider equation (4.1) from the perspective of deriving multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, λ-symmetries, null forms and integrating factors sequentially, and demonstrate the effectiveness of exploring the interconnections. From these quantities we also derive integrals and the general solution of this equation for the sake of completeness.

We begin our analysis with Lie point symmetries. Equation (4.1) admits a set of three-parameter Lie point symmetries of the form
*D*[*Q*]/*Q*, we can derive the λ-symmetries associated with the above vector fields, which in turn read
*U* and *S*. To obtain them, we determine the function *V* using the relation (3.13). We obtain
*X* from the known expression of *V* through the relation (3.2). The resultant expressions read
*X* and *V*, we can obtain the null forms *U* and *S*, respectively (see equation (3.1)), as
*U* and *S* in equation (2.27) and solving the resultant equation, we find the integrating factors for (4.1) in the form
*R*_{1},*R*_{2} and *R*_{3} also become adjoint-symmetries of (4.1), which can be directly verified by substituting them in (2.15). Using the expressions for *R* and *V*, we can fix the Darboux polynomials for the given equation by recalling the relation *F*=*V*/*R*. The Darboux polynomials turn out to be
*F*_{11},*F*_{12} and *F*_{13} share the common cofactor *x*+2*D*[*F*]=(*x*+2*F* as
*F*_{3}/*F*_{1} and *F*_{2}/*F*_{1} lead to the first integrals *I*_{1} and *I*_{2} (see equation (4.12)), respectively. The JLM associated with the given equation can be obtained from the Darboux polynomials (4.10) by recalling the relation *F*=*M*^{−1}.

Once we know the null forms and integrating factors, we can construct the associated integrals of motion by evaluating the integrals given in (2.31). Our analysis shows that
*I*_{1}, *I*_{2} and *I*_{3} are the three independent integrals for the given equation (4.1). From our knowledge of *I*_{1}, *I*_{2} and *I*_{3}, we get the general solution in the form

### (b) Example 2

We consider another interesting example [10]
*et al.* [20] and they derived the null forms, integrating factors and integrals through this method. The exact expressions are given in table 2. The procedure given in the previous example may also be followed for this example to derive the quantities displayed in table 2. Here also because of non-uniqueness of *S*, *U* with λ, one may find two independent integrals, *I*_{2} and *I*_{3}, from the same λ-function. The vector field *Ω*_{1} provides one independent integral, *I*_{1}, and the vector field *Ω*_{2} gives the two independent integrals, *I*_{2} and *I*_{3}. The vector field *Ω*_{3} provides only a dependent integral.

Using the integrals *I*_{1},*I*_{2} and *I*_{3} the solution of equation (4.14) can be written down implicitly as

### (c) Example 3

Finally, we consider the following example:
*S*_{i} and *U*_{i}, *i*=1,2,3:
*S* and *U*, we can deduce the λ-symmetries by solving equation (3.11). Doing so, we find
*S*_{3},*U*_{3}) provides another λ-symmetry which is given by
_{1}(=λ_{2}) and *V* and *X* can be determined with the help of equation (3.1) as
*V* and *R* are known now, the Darboux polynomials admitted by equation (4.16) are found by exploiting the relation (3.4). We observe
*F*=*M*^{−1}. Jacobi last multiplier are given by

## 5. Panorama of interconnections

In the previous section, starting from Lie point symmetries we derived all other quantities. Since the interconnections are global one can consider any other quantity in this family, and derive the rest of them; see the connection diagram (figure 1). In this section, we consider two such cases, namely (i) integrating factors as starters and (ii) Darboux polynomials as starters, and demonstrate the method of deriving all other quantities from them.

### (a) From integrating factors to others

In this section, we demonstrate that starting from the integrating factor we can derive all other quantities for the example (4.1). Suppose an integrating factor *R*_{1} (*R*_{1}=−1/*x**U*_{1} from equation (2.27), which exactly matches with the first expression given in equation (4.8). Substituting *U*_{1} in (3.1) and rewriting the resultant equation, we obtain *D*[*V* _{1}]−(*V* _{1}=0. A particular solution to this equation is *V* _{1}=−*V* _{1}, we can get an expression for *X*_{1} through the relation equation (3.2), which also exactly matches with the first expression given in (4.6). Once we have *X*_{1} and *V* _{1} we can obtain the second null form *S*_{1} with the help of equation (3.1). Once we know *S*_{1} and *U*_{1}, we can construct the λ-symmetry through the expression (3.11), i.e. *D*[λ]+λ^{2}−(*x*=0. A particular solution to this equation is λ=*x*. Using the quantities *R*_{1} and *V* _{1} in (3.4), we can find the Darboux polynomials as given in the first expression in (4.10). In this way, we can derive the rest of the quantities from an integrating factor. The exact expressions for the other quantities can be found in table 1. The associated integral turns out to be *I*_{1} as expected. The procedure is exactly the same for the other two integrating factors *R*_{2} and *R*_{3}, which are given in (4.9). The integrating factors *R*_{2} and *R*_{3} provide exactly the same expressions (*S*_{2},*U*_{2},*R*_{2}) and (*S*_{3},*U*_{3},*R*_{3}). We also note here that if we start from an integrating factor other than the above three then proceeding in the same manner as outlined above one can get their associated null forms and Darboux polynomials. However, these expressions do not lead to any new integrals.

### (b) From Darboux polynomials to others

Suppose Darboux polynomials which share the same cofactor are given as the first information and we have to determine the rest of the quantities. From the Darboux polynomials, we can find the integrals using the ratios. We assume that the two Darboux polynomials *F*_{12}=*x*^{2} and *F*_{3}=*I*_{1}=*x**R*_{1} given in (4.9). The method of deriving the rest of the quantities from *R*_{1} is outlined in the previous sub-section. In this way, from Darboux polynomials one can derive the rest of the quantities. The procedure is the same for the other Darboux polynomials.

One may assume that the last multipliers are given and then consider the task of determining the other quantities. Using the relation *F*=*M*^{−1}, we can find the Darboux polynomials. Once Darboux polynomials are known the procedure given in §4*c* may be followed to derive the rest of the quantities.

## 6. Conclusion

In this work, we have demonstrated the interconnection between the Prelle–Singer method (or any one of the methods studied in this paper as the starting point) and the other existing well-known methods in the literature such as JLMs, Darboux polynomials, Lie point symmetries, adjoint-symmetries and λ-symmetries in the case of third-order nonlinear ODEs. For this purpose, we started with the Prelle–Singer method. In the Prelle–Singer method, the quantities, namely (i) null forms *U* and *S* and (ii) integrating factors *R* which are determined by three first-order equations, play a major role. By introducing suitable transformations to the null forms *U* and *S* and to the integrating factor *R*, we have related these three quantities to the above other quantities. While relating the Prelle–Singer method with Darboux polynomials and the JLM, we introduced the transformation *V*/*F* in the integrating factor equation. To relate the adjoint-symmetries with the integrating factor, we rewrite the three first-order equations in the Prelle–Singer method in terms of a single variable *R*. We then demonstrate that this third-order equation in the variable *R* coincides with the adjoint-symmetry equation. The difficult and unknown connection between λ-symmetries and the null forms has been demonstrated by using the compatibility between the λ-determining equation and the determining equation for the null forms in the Prelle–Singer method. We have recalled the known connection λ=*D*[*Q*]/*Q* to relate the Lie point symmetries and the null forms. In this way, we have established the interconnections between the null forms, integrating factors, adjoint-symmetries, λ-symmetries, Lie point symmetries, Darboux polynomials and JLMs. We have observed that some of the connections are common for both second-order as well as third-order ODEs, while the others are specifically applicable to third-order ODEs. We have illustrated these interconnections with three definitive examples discussed in the literature. Currently, we are extending the above procedure to *n*th-order nonlinear ODEs. We have obtained some interesting results in this direction. The results will be published in the near future.

## Authors contributions

All the authors have contributed equally to the research and to the writing up of the paper.

## Funding statement

R.M.S. acknowledges the University Grants Commission (UGC-RFSMS), Government of India, for providing a Research Fellowship. The work of M.S. forms part of a research project sponsored by the Department of Science and Technology, Government of India. The work of M.L. is supported by a Department of Science and Technology (DST), Government of India, IRHPA research project. M.L. is also supported by a DAE Raja Ramanna Fellowship and a DST Ramanna Fellowship programme.

## Competing interests

We have no competing interests.

- Received September 23, 2014.
- Accepted February 20, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.