## Abstract

In this work, we establish a connection between the extended Prelle–Singer procedure and other widely used analytical methods to identify integrable systems in the case of *n*th-order nonlinear ordinary differential equations (ODEs). By synthesizing these methods, we bring out the interlink between Lie point symmetries, contact symmetries, λ-symmetries, adjoint symmetries, null forms, Darboux polynomials, integrating factors, the Jacobi last multiplier and generalized λ-symmetries corresponding to the *n*th-order ODEs. We also prove these interlinks with suitable examples. By exploiting these interconnections, the characteristic quantities associated with different methods can be deduced without solving the associated determining equations.

## 1. Introduction

In two of our earlier works [1,2], we have interconnected six widely used analytical methods for solving ordinary differential equations (ODEs), namely the extended Prelle–Singer method, Lie symmetry analysis, the Jacobi last multiplier method, the Darboux method, the adjoint-symmetries method and the λ-symmetries approach by considering second- and third-order ODEs. Progressing further, in this paper, we unearth the interconnection between the extended Prelle–Singer procedure with two more procedures, namely (i) contact symmetries and (ii) generalized λ-symmetries. By establishing these new interconnections we are able to bring the eight different analytical methods under one umbrella. More importantly, we prove the presence of the interconnections up to general *n*th-order ODEs.

Some of the results we report here are new to the literature. We also note here that the first attempt to connect the Prelle–Singer procedure with other methods, in particular λ-symmetries in the second-order ODEs, came from Muriel & Romero [3]. They have shown that λ-symmetries are nothing but the null forms (with a negative sign) given in the Prelle–Singer procedure. In this work, we study this interconnection between the null forms in the extended Prelle–Singer procedure and λ-symmetries to higher orders. The other important result which we report in this paper is the interconnection between generalized λ-symmetries and the extended Prelle–Singer method quantities, namely null forms and the integrating factor. It is very difficult to determine both of them by solving their determining equations. However, using the proposed interconnections, one can obtain these symmetries in a simple and straightforward manner.

We demonstrate all these interconnections with suitable examples in all orders. For example, in second- and third-order ODEs, we recall the same examples which we considered earlier and demonstrate the new interconnections so that the wider interconnection can now be appreciated. To demonstrate the validity of the interconnections in higher order, we consider an example also from fourth order. The realization of the interplay at the *n*th order is also demonstrated through the *n*th-order free particle equation.

Some of the interconnections are not mere extensions of the interconnections from the case of second- and third-order ODEs. The interconnections are achieved by introducing suitable transformations in the Prelle–Singer procedure quantities, namely the null forms, *S*_{i}, *i*=1,2,…,*n*−1, and the integrating factor, *R*. These interconnections will be briefly explained in the following sections.

The plan of the paper is as follows. In §2, we give the definition and the determining equations of the Prelle–Singer procedure for solving *n*th-order ODEs. In addition, we discuss the several well-known methods such as Lie point symmetries, contact symmetries, λ-symmetries, generalized λ-symmetries, the Jacobi last multiplier, Darboux polynomials and adjoint-symmetries methods and their determining equations. We also discuss the known and unknown interconnections among all these methods. In §3, we show the interconnections among all these methods with examples. Finally, we summarize our results in §4.

## 2. Interconnections

To prove the interconnections among all the above methods, we start our discussion with the extended Prelle–Singer method. In the following, we discuss all the other methods in relation to the extended Prelle–Singer method.

Consider an *n*th-order ODE of the following form:
*ϕ* is the function of *t*,*x*,*x*^{(1)},…,*x*^{(n−1)}.

### (a) Extended Prelle–Singer method

In 1983, Prelle & Singer [4] proposed a procedure for solving first-order ODEs that presents the solution, if such a solution exists, in terms of elementary functions. Subsequently, Duarte *et al.* [5] extended the underlying ideas to second-order ODEs and constructed only one integral for a class of equations. Later three of the present authors extended the algorithm given by Duarte *et al.* in such a way that the extended algorithm will provide two independent integrals for the given second-order ODE [6]. The same authors have extended the algorithm to the third order and *n*th order as well as coupled ODEs and established that the extended Prelle–Singer procedure is a stand-alone method to determine integrating factors, integrals and the general solution of ODEs of any order including coupled ones, provided they are integrable. For more details about this method, one may refer to Chandrasekar *et al*. [6–8].

In the following, we recall briefly the extended Prelle–Singer procedure applicable for *n*th-order ODEs.

### Definition 2.1

Consider an *n*th-order ODE (2.1). Let the ODE (2.1) admit a first integral *I*(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)})=*C*, where *C* is a constant. Let *S*_{i}, *i*=1,2,…,*n*−1, and *R* denote the null forms (which are essentially functions) and the integrating factor. They can then be determined from the relations [8]
*a*
*b*
*c*
*d*
*e*
*f*
*g*
*h*
where the total differential operator *D* is defined by *S*_{i}(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)})*x*^{(i)} *dt*−*S*_{i}(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)}) *dx*^{(i−1)}=0, *i*=1,2,…,*n*−1.

Once we know the null forms *S*_{i}, *i*=1,2,…,*n*−1, and the integrating factor *R*, we can construct the integrals of the ODE (2.1) through the expression [8]
*I*_{x}=−*RS*_{1}, *I*_{x(i)}=−*RS*_{i+1}, *i*=1,2,…,*n*−2, *I*_{x(n−1)}=−*R*. Once we know *n* independent integrals, we can derive the general solution of the given *n*th-order ODE from these integrals.

#### (i) Transformations

In the extended Prelle–Singer method, the null forms *S*_{i}, *i*=1,2,…,*n*−1, and integrating factor *R* play a major role in finding the integrals.

To connect the null forms and integrating factors with other integrability quantifiers, we introduce the following transformations in *S*_{i} and *R*, that is:
*a*and
*b*where *V* (*t*,*x*,…,*x*^{(n−1)}) and *X*_{i}(*t*,*x*,…,*x*^{(n−1)}) are *n*−1 unknown functions and *D* is the total differential operator. With this substitution, equations (2.2c), (2.2a) and (2.2b) now become a system of linear equations, respectively:
*a*
*b*
*c*We introduce yet another transformation,
*F*(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)}) is a function to be determined in (2.2d) so that the latter equation can be rewritten in a compact form in the new variable *F* as
*F*. The transformations given in equations (2.4) and (2.6) interlink the null forms and integrating factor with other integrability quantifiers, say λ-symmetries, Lie point symmetries, contact symmetries, Darboux polynomials, the Jacobi last multiplier, adjoint symmetries and generalized λ-symmetries, as we see below.

### (b) Lie point symmetry analysis

Lie point symmetry analysis is one of the powerful methods of deriving classes of solutions of differential equations in an algorithmic way. The underlying idea here is to enforce the given equation to be form invariant under an infinitesimal transformation involving independent and dependent variables. The infinitesimal transformations, which leave the given equation form invariant, are in turn called Lie point symmetries [9,10]. Our primary interest here is to explore how these Lie point symmetries are intrinsically linked with the other integrability quantifiers.

### Definition 2.2

The invariance of equation (2.1) under a one-parameter group of Lie point symmetries, corresponding to the infinitesimal transformations
*ξ*(*t*,*x*) and *η*(*t*,*x*) are functions of their arguments and *ε* is a small parameter, demands the following condition to be satisfied [9]:
*η*^{(j)} is the *j*th prolongation, *j*=1,2,3,…,*n*, of the infinitesimal point transformations (2.8) and is defined as
*t*.

Substituting the known expression *ϕ* in (2.9) and solving the resultant equation we can obtain the Lie point symmetries associated with the given *n*th-order ODE. The associated vector field is given by

One may also introduce the characteristics
*Q* in the form
*ξ* and *η* associated with the Lie point symmetries from out of the class of solutions to (2.12), which depend only on *x* and *t*, and also have a linear dependence in *x*^{(1)}.

### Theorem 2.3

*The connection between null forms in the extended Prelle–Singer procedure and Lie point symmetries is given by
**where D represents the total derivative operator.*

### Proof.

Let *I*(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)}) be the first integral of the given *n*th-order ODE (2.1). Then
*Q*(=*η*−*x*^{(1)}*ξ*) and using the Prelle–Singer procedure quantities *S*_{i} and *R* as *I*_{x}=−*RS*_{1}, *I*_{x(i)}=−*RS*_{i+1}, *i*=1,2,…,*n*−2, *I*_{x(n−1)}=−*R*, we can obtain the expression
*n*=2, the above relation reads *D*[*Q*]+*S*_{1}*Q*=0. Likewise, the above relation for the third-order ODEs turns out to be *D*^{2}[*Q*]+*S*_{2}*D*[*Q*]+*S*_{1}*Q*=0. ▪

#### (i) Other quantifiers from Lie point symmetries

Substituting the transformations (2.4a,*b*) in (2.13), we can rewrite the latter equation in terms of *V* and *X*_{i}, which upon solving yields these quantities. Substituting them back in equations (2.4a,*b*), we can obtain the null forms *S*_{i} of the given equation. The associated integrating factor can be obtained by solving equation (2.2d). As we demonstrated in our earlier works, the null forms and the integrating factor *R* can be connected to adjoint symmetries, the Jacobi last multiplier, Darboux polynomials and λ-symmetries, and we can relate all these quantifiers recursively. Once we know the Lie point symmetries, λ-symmetries can be determined from (2.27), which is given below. From the null forms and the integrating factor, we can find the generalized λ-symmetries using the relation (2.52), which is given in §2h.

### (c) Contact symmetries

Several nonlinear ODEs do not admit Lie point symmetries but are proved to be integrable by other methods. To demonstrate the integrability of these nonlinear ODEs in the sense of Lie, one should consider more generalized transformations. One such transformation includes velocity-dependent (first derivative) terms in the infinitesimal transformations [10]. In the following, we give a brief account of the velocity-dependent transformations and how they can be related to null forms and integrating factors that appear in the Prelle–Singer procedure. The last result is new to the literature.

Let us consider a one-parameter group of contact transformations [10]
*ξ*(*t*,*x*,*x*^{(1)}), *η*(*t*,*x*,*x*^{(1)}) and *η*^{(1)}(*t*,*x*,*x*^{(1)}) are functions of their arguments and *ε* is a small parameter. The functions *ξ* and *η* determine an infinitesimal contact transformation if it is possible to write them in the form [11]
*W*(*t*,*x*,*x*^{(1)}) is an arbitrary function of its arguments. If *W* is linear in *x*^{(1)} the corresponding contact transformation is an extended point transformation and it holds that *W*(*t*,*x*,*x*^{(1)})=*η*(*t*,*x*)−*x*^{(1)}*ξ*(*t*,*x*). The invariance of equation (2.1) under the infinitesimal contact transformation is given by
*η*^{(j)} is the *j*th prolongation, *j*=1,2,3,…,*n*, of the infinitesimal transformation (2.17). The prolongations are defined as
*t*. The associated contact symmetry vector field is given by *Ω*=*ξ*(*t*,*x*,*x*^{(1)})(∂/∂*t*)+*η*(*t*,*x*,*x*^{(1)})(∂/∂*x*).

Analogous to the case of Lie point symmetries (see equation (2.11)), one may introduce the characteristics [10]
*W* in the form
*W*. From *W*, one can recover the contact symmetries *ξ* and *η*. However, unlike the Lie point symmetries, it is very difficult to determine them systematically.

### Theorem 2.4

*The connection between the null forms and the contact symmetries is given by
**where D represents the total derivative operator.*

### Proof.

The proof is analogous to the one which we discussed in the earlier section, that is, the connection between the Lie point symmetries and the Prelle–Singer procedure. Here we consider the contact symmetry characteristics *W* instead of the Lie point symmetry characteristics *Q*. ▪

### (d) λ-symmetries

As noted above, the conventional Lie point symmetry analysis has been generalized in several directions such that the nonlinear ODEs which cannot be solved by point symmetries can now be integrated with the help of suitable generalized transformations. Another such generalized symmetry is the λ-symmetry [3,12–15]. The components of these vector fields must satisfy a system of determining equations that depend on an arbitrary function λ, which can be chosen to solve the system easily. When this arbitrary function is chosen to be null, we obtain the classical Lie point symmetries. This method also provides a systematic procedure to find the first integrals and the integrating factors.

### Definition 2.5

Consider an *n*th-order ODE which admits a λ-symmetry *t*,*x*,*x*^{(1)},…,*x*^{(n−1)}), then the invariance of the *n*th-order ODE under the λ-symmetry vector field is given by [12]
*ξ*(∂/∂*t*)+*η*(∂/∂*x*)+*η*^{[λ,(1)]}(∂/∂*x*^{(1)})+⋯+*η*^{[λ,(n)]}(∂/∂*x*^{(n)}). Here *η*^{[λ,(1)]}, *η*^{[λ,(2)]},…,*η*^{[λ,(n)]} are the first, second, …, *n*th λ-prolongations, respectively, whose explicit expressions are given by [12]

The λ-prolongation (2.25) reduces to the classical Lie point prolongation formula (2.9) when λ=0. Solving the invariance condition (2.26), we can obtain the explicit forms of *ξ*,*η* and λ.

Suppose the given ODE admits Lie point symmetries, then the λ-symmetries can be derived without solving the invariance condition (2.26). The λ-symmetries can be directly obtained from the Lie point symmetries through the expression [3]
*D* is the total differential operator and *Q*=*η*−*x*^{(1)}*ξ* provided the expression given in (2.27) satisfies (2.26). The associated λ-symmetry vector field is given by *n*th-order ODE (2.1) by solving the invariance condition (2.26).

#### (i) Connection between λ-symmetries and the Prelle–Singer method

The interconnection between λ-symmetries and the Prelle–Singer method for the second- and third-order ODEs was discussed elaborately in [1–3]. Here, we generalize the interlink to *n*th-order ODEs.

### Theorem 2.6

*The null forms associated with the nth-order ODE (*2.1*) in the extended Prelle–Singer procedure can be connected to the λ-symmetries through the relation
**with the assumption that S*_{n}*=1.*

### Proof.

Let *I*(*t*,*x*,…,*x*^{(n−1)}) be the first integral of (2.1), then *R*=−*I*_{x(n−1)} is an integrating factor. The total derivative d*I*/d*t*=0 gives
*I*(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)}) also be a first integral of *t*,*x*,*x*^{(1)},…,*x*^{(n−1)}), then
*I*_{x}=−*RS*_{1}, *I*_{x(i)}=−*RS*_{i+1}, *i*=1,2,…,*n*−2, *I*_{x(n−1)}=−*R* from the Prelle–Singer method and substituting them in (2.32) and rewriting the later we will end up with (2.28). ▪

The above result shows that in the case of second-order ODEs, since we have only one null form *S*_{1}, it is directly connected with the λ-symmetry through the expression λ=−*S*_{1} [1,3]. In the case of third-order ODEs, we have two null forms, namely *S*_{1} and *S*_{2}, which can be connected to the λ-symmetries through the differential relation *D*[λ]+λ^{2}+*S*_{2}λ+*S*_{1}=0 [2]. From (2.28), we infer that, in fourth-order ODEs, λ-symmetries and null forms are connected by the differential relation (*D*^{2}[λ]+3*λD*[λ]+λ^{3})+(*D*[λ]+λ^{2})*S*_{3}+*λS*_{2}+*S*_{1}=0. The interconnection persists in all orders and for an *n*th-order ODE the explicit expression is given by (2.28). We note here that while deriving (2.28) we assumed that λ≠0. In the case λ=0, we have *I*_{x}=0 (see equation (2.32)).

#### (ii) Other quantifiers from λ-symmetries

From the known λ-symmetries and by substituting the transformations (2.4a,*b*) in expression (2.28), we can rewrite the latter equation in terms of *V* and *X*_{i}, which upon solving yields these quantities. Substituting them back in equations (2.4a,*b*), we can obtain the null forms *S*_{i} of the given equation. From the null forms, we can obtain the integrating factor through (2.2d). From the Prelle–Singer method quantities, we can identify the rest of the integrability quantifiers.

### (e) Jacobi last multiplier method

The Jacobi last multiplier method is yet another important analytical method to prove the integrability of the given dynamical system [16,17]. This method helps to determine the integrals associated with the given equation. The multipliers can also be used to find the Lagrangians of the associated ODE whenever the considered ODE is of even order [18].

### Definition 2.7

Let us assume that the ODE (2.1) admits a first integral *I*(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)})=*M*_{1}/*M*_{2}=*C*, where *C* is a constant on solutions, and *M*_{i}, *i*=1,2, are the associated Jacobi last multipliers. These multipliers can be determined for the given *n*th-order ODE by solving the following determining equation [18]:
*D* represents the total differential operator. Substituting the given equation in (2.33) and solving the resultant equation, we can obtain the last multipliers associated with an *n*th-order ODE. The ratio of any two Jacobi last multipliers, say *M*_{1}/*M*_{2}, gives the first integral. Note that *M*_{1} and *M*_{2} may also be trivially related, that is, *M*_{1}=*αM*_{2}, where *α* is a constant number.

In the following subsections, the connection between the Jacobi last multiplier, Lie point symmetries and the Prelle–Singer method is discussed.

#### (i) Jacobi last multiplier and Lie point symmetries

The connection between Lie point symmetries and the Jacobi last multiplier has been known for a long time [18]. It is given by
*ξ*_{1},*η*_{1}), (*ξ*_{2},*η*_{2}),…,(*ξ*_{n},*η*_{n}) are the Lie point symmetries of the *n*th-order ODE, *i*=1,2,…,*n*−1, are their corresponding prolongations, respectively, and the inverse of Δ defines the multiplier of the given equation provided that Δ≠0. Note that there are cases where Jacobi last multipliers exist even without Lie point symmetries. In such cases, the existence of (2.34) may not be valid. However, the other quantifiers can be determined as pointed out in §2e(iii).

#### (ii) Jacobi last multiplier and Prelle–Singer method

The connection between the Jacobi last multiplier and the extended Prelle–Singer method is shown below.

### Theorem 2.8

*The connection between the integrating factor R of the extended Prelle–Singer procedure and the Jacobi last multiplier M is given by
**where V is the function defined by (*2.4*).*

### Proof.

By comparing equation (2.33) with (2.7), we observe that

Since *R* and *V* are known from the Prelle–Singer method, the Jacobi last multiplier can be derived from the Prelle–Singer method itself. This establishes the connection between the Prelle–Singer method and the Jacobi last multiplier method. ▪

#### (iii) Other quantifiers from Jacobi last multipliers

Suppose that if we know the Jacobi last multipliers, then the ratio of the multipliers gives the first integral. Once we know the integral, we can identify the integrating factor *R* through the relation *R*=−*I*_{x(n−1)}. Once we know the integrating factor, we can obtain the null forms *S*_{i} from equations (2.2*a*–*c*). We can determine the λ-symmetries from the null forms through the relation (2.28). The rest of the quantities can be constructed as outlined in §2d(ii).

### (f) Darboux polynomials approach

The Darboux theory of integrability is yet another approach which helps to determine the integrals of the given ODE. The ratio of two Darboux polynomials gives an integral provided they have the same cofactor [19]. The Darboux theory is extensively studied in the contemporary literature [20].

### Definition 2.9

Let us assume that the ODE (2.1) admits a first integral *I*(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)})=*C*, where *C* is a constant on solutions. The Darboux polynomial determining equation for an *n*th-order ODE (2.1) is given by [19]
*D* is the total differential operator and *g*(*t*,*x*,…,*x*^{(n−1)}) is the cofactor. Solving equation (2.39) through the appropriate ansatz on *f* and *g*, we can obtain the Darboux polynomials (*f*) and the cofactor *g*. The ratio of two Darboux polynomials *f*_{1}/*f*_{2} which share the same cofactor *g* gives a first integral.

The combinations of the Darboux polynomials also define a first integral. For example, let us consider the function *f*_{i}′s are the Darboux polynomials and *n*_{i}′s are rational numbers. If we can identify a sufficient number of Darboux polynomials, *f*_{i}′s, satisfying the relations *D*[*f*_{i}]/*f*_{i}=*g*_{i}, where *g*_{i}’s are the co-factors, then

### Corollary 2.10

*The connection between the integrating factor R of the extended Prelle–Singer procedure and the Darboux polynomials is given by*
*provided the cofactor of the Darboux polynomial is ϕ*_{x(n−1)}.

### Proof.

Fixing *G*=*F* in equation (2.40) and comparing the latter equation with equation (2.7), we find that the cofactor of the Darboux polynomial equation is
*G*=*F* with the cofactor *ϕ*_{x(n−1)} in equation (2.6), we obtain *R*=*V*/*F*. ▪

#### (i) Connection between Darboux polynomials and Jacobi last multiplier

By comparing equations (2.39) and (2.33), we can identify the connection between the Darboux polynomials and the Jacobi last multiplier, provided its cofactor *g*=*ϕ*_{x(n−1)}, as
*n*th-order ODEs, provided the cofactor of Darboux polynomials is *ϕ*_{x(n−1)} [1].

#### (ii) Other quantifiers from Darboux polynomials

Darboux polynomials are given as starters, then the ratio of the Darboux polynomials gives the first integral. Once we know the integral, we can identify the integrating factor *R* through the relation *R*=−*I*_{x(n−1)}. From the known integrating factor, we can obtain the null forms *S*_{i} from equations (2.2*a*–*c*). From the Prelle–Singer method quantities, we can find the other integrability quantifiers.

### (g) Adjoint symmetries

An integrating factor is a set of functions that on multiplying the given ODE yields a first integral [10]. If the system is self-adjoint, then its integrating factors are necessarily solutions of its linearized system (2.12). Such solutions are the symmetries of the given system. If the given ODE is not self-adjoint, then its integrating factors are necessarily solutions of the adjoint system of its linearized system. Such solutions are known as adjoint symmetries of the given ODE [10,21].

### Definition 2.11

The adjoint ODE of the linearized symmetry condition (2.12) can be written as
*D* and *Λ* represent the total derivative operator and adjoint symmetries, respectively. Solutions of equation (2.44) are called adjoint symmetries [21].

If these solutions also satisfy the adjoint-invariance condition

#### (i) Adjoint symmetries and Prelle–Singer method

The connection between adjoint symmetries and the Prelle–Singer method is given below.

### Theorem 2.12

*The connection between adjoint symmetries and the Prelle–Singer method is given by
*

### Proof.

Rewriting equations (2.2a–*d*) as a single equation in one variable, *R*, we find
*R* is nothing but the adjoint symmetry *Λ*, that is,
*n*th-order nonlinear ODE. ▪

#### (ii) Other quantifiers from adjoint symmetries

Since adjoint symmetries are also integrating factors provided they satisfy equation (2.45), we can find the null forms from them through the relations (2.2a–*c*). From the null forms, we can find the λ-symmetries using equation (2.28). From the λ-symmetries, the expressions *V* and *X*_{i} can be obtained from the null forms by using expressions (2.4a,*b*). We can also obtain the Lie point symmetries with the help of (2.16). From a knowledge of the integrating factor *R* and the function *V* , the Darboux polynomials can be determined from (2.6). Then the Jacobi last multiplier can also be found from the inverse of the Darboux polynomials.

### (h) Generalized λ-symmetries

Generalized λ-symmetries contain the well-known subclasses of vector fields that have appeared in the literature. The coefficient functions in the infinitesimal generators are not only functions of the dependent and independent variables, but also functions of their derivatives up to the order *n*−1 and an arbitrary function λ. For more details on these generalized λ-symmetries, one may refer to Muriel & Romero [13].

### Definition 2.13

The generalized vector field is given by
*ξ*(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)}), *η*(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)}) and *ζ*^{(1)}(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)}) are the three arbitrary functions which we have to determine. Their prolongations are given by the following expressions [13]:
*i* takes a value from 2 to *n*. Suppose *ξ*, *η* and *ζ*^{(1)} are functions of *t*, *x* and *x*^{(1)} alone, then the generalized λ-symmetries are called telescopic vector fields [22]. We note here that the function λ is given by λ=(*ζ*^{(1)}+*x*^{(1)}*D*[*ξ*]−*D*[*η*])/(*η*−*x*^{(1)}*ξ*).

### Theorem 2.14

*The generalized λ-symmetries are related to the integrating factors and null forms of the Prelle–Singer method through the expression
**where, for i*+1=*n, we assume that S*_{n}=1. *The underlying generalized* λ-*symmetry vector field reads*

### Proof.

Let *I*(*t*,*x*,*x*^{(1)},…,*x*^{(n−1)}) be the first integral of the given *n*th-order ODE, then
*ξ*=0 and *η*=*R* and substitute the expressions *I*_{x}=−*RS*_{1}, *I*_{x(i)}=−*RS*_{i+1}, *i*=1,2,…,*n*−2, *I*_{x(n−1)}=−*R* (which appear in the Prelle–Singer procedure) in (2.55) and rewrite the latter expression suitably to obtain equation (2.52). The generalized λ-symmetries turn out to be of the form

From the generalized λ-symmetries, we can find the integrating factors and null forms using the relation (2.52). Once we know the null forms and integrating factors we can determine all other quantifiers as we described earlier.

The broader interconnections which exist between the eight different analytical methods in the literature are summarized in figure 1. As we pointed out earlier in the extended Prelle–Singer procedure, we have two sets of quantities, namely the null forms and integrating factors, which determine the integrability. How these two quantities are interrelated with other quantifiers is schematically given in figure 1. The null forms *S*_{i} can be connected through Lie point symmetries, contact symmetries (where *Q* is replaced by *W*) and λ-symmetries. On the other hand, the adjoint symmetries, Jacobi last multiplier and Darboux polynomials can be brought out from the integrating factor. Generalized λ-symmetries can be related to the Prelle–Singer procedure through the null forms and integrating factor. We wish to note here that, in the cases where no Lie point symmetries exist, part of figure 1 alone will be absent. Other interconnections between the integrability quantifiers will remain the same as in figure 1.

## 3. Examples

In this section, we prove the above interconnections by considering an example in each of the second-, third-, fourth- and *n*th-order differential equations. As we generalize our earlier results, we do not discuss the method of finding all the quantities for second- and third-order ODEs. Instead, for these two cases, we report the method of finding contact symmetries and generalized λ-symmetries from the Prelle–Singer procedure which we have not discussed earlier. The interconnections for the higher order ODEs (*n*>3) are demonstrated explicitly.

### (a) Second-order ordinary differential equations

We start our illustration with the same second-order nonlinear ODE as we considered in our earlier paper, namely [1]
*W*=*η*−*x*^{(1)}*ξ*, we can determine the contact symmetries of (3.1). The associated vector field reads
*ζ*^{(1)}=−*RS*)

### (b) Third-order ordinary differential equations

To demonstrate the interconnection in third-order ODEs, we again consider the same example as we considered in our earlier work [2]; that is,
*W*=*η*−*x*^{(1)}*ξ*, we obtain the contact symmetries as (*ξ*=0 for the first two symmetries, *η*=0 for the third symmetry)
*ζ*^{(1)} can be obtained by substituting the null forms and the integrating factors in equation (2.52). The latter equation turns out to be
*R* and using the expressions (2.51), we can obtain the other components *ζ*^{(2)} and *ζ*^{(3)} in the generalized λ-symmetries. The generalized λ-symmetries are given by

### (c) Fourth-order ordinary differential equations

The interconnections in fourth-order ODEs are yet to be demonstrated in the literature. So we intend to consider an example which is governed by the fourth-order ODE, namely [10]

To obtain the null forms in the Prelle–Singer procedure, we have to solve the determining equations (2.2a–*c*). Doing so, we find

We will use *S*_{3i} to establish the integrability of the given equation. The rest of the null forms will be used to demonstrate the interconnections. From the null forms *S*_{3i}, *i*=1,2,3,4,5, with equation (2.2d), we can obtain the integrating factors as
*a*and
*b*The null forms (3.24) and the integrating factors (3.25) should satisfy the constraints given in equations (2.2e–*h*). Since we are dealing with a fourth-order ODE, the above forms can lead to the four independent integrals. Their forms are given below.

From the null forms *S*_{1},*S*_{2} and *S*_{3}, we can obtain the λ-symmetries by solving the relation (2.28). The resultant expressions read
*S*_{12},*S*_{22} and *S*_{32} provide another λ, that is,

In the following, we illustrate the interconnections by considering the vector fields

From the λ-symmetries, we can find the functions *V*, *X*_{1} and *X*_{2}, which are of the form (see equations (2.5a–*c*))
*a*
*b*
*c*
*d*

From the integrating factors and the quantity *V* , we can find the Darboux polynomials associated with equation (3.23) through the relation *F*=*V* _{i}/*R*_{i}, *i*=1,2,3,4. The obtained Darboux polynomials are given by
*ζ*^{(1)} of the generalized vector field using equation (2.52). The resultant expressions read

From the null forms (3.24) and the integrating factors (3.25b), we can find the integrals using the relation (2.3) admitted by equation (3.23). The integrals are found to be

### (d) *n*th-order ordinary differential equations

To verify the validity of our results in the case of an *n*th-order ODE, we consider the following example:
*n*th-order.

In the first column of table 1, we present the Lie point symmetries of an *n*th-order ODE. From the Lie point symmetries, we proceed to construct λ-symmetries, null forms, integrating factors, Darboux polynomials, the first prolongation component of the generalized λ-symmetries and finally the integrals for the *n*th-order ODE through the procedure outlined in the previous section. The explicit expressions are all given in table 1. We have verified that all of these quantifiers satisfy their respective determining equations. The results support that the theory developed in this paper is applicable to *n*th-order ODEs.

We mention here that the second-order free particle equation admits eight Lie point symmetries. However, in table 1, one may observe that, when *n*=2 (second-order ODE), one can obtain six Lie point symmetry generators only. The reason for this is that the six generators which are given in table 1 appear in every order starting from second order and so we have generalized these six symmetry vector fields. The other two vector fields *x*(∂/∂*t*) and *xt*(∂/∂*t*)+*x*^{2}(∂/∂*x*) of the second-order ODE do not appear in the higher order. Since these two vector fields are missing in all higher orders we have not included them in table 1.

## 4. Conclusion

In this work, we have developed a systematic procedure to make interconnections among several analytical methods which are available in the literature for *n*th-order nonlinear ODEs. In the Prelle–Singer procedure, the important quantities are the null forms *S*_{i}, *i*=1,2,3,…,*n*−1, and the integrating factor *R*. By introducing suitable transformations in the null forms *S*_{i} and the integrating factor *R*, we can relate the other methods such as λ-symmetries, Lie point symmetries, the Jacobi last multiplier, Darboux polynomials, the adjoint-symmetries method and generalized λ-symmetries with the Prelle–Singer procedure. The λ-symmetries, Lie point symmetries and generalized λ-symmetries are connected with the Prelle–Singer procedure through null forms *S*_{i}. The other methods, the Jacobi last multiplier, Darboux polynomials and the adjoint symmetries, are related to the Prelle–Singer method by the integrating factors and the expression *V* , which can be obtained from the null form *S*_{n−1}. By using these interconnections, one can find the relevant quantities associated with the other methods without solving their own determining equations. We have explained the above interconnections with several examples, that is, second-, third-, fourth- and *n*th-order ODEs.

Finally, we would to like to point out that the interconnections which we have presented in the manuscript will also work for partially integrable systems (that is, a lesser number of integrals than the required *n* independent integrals). For example, if we know one integral, we can obtain the quantities associated with the other methods corresponding to this integral using the interconnections which we have discussed above.

Also, there may be situations where the above interconnections are difficult to achieve. For example, consider the second-order ODE *x*^{(2)}+*t*^{2}/4*x*^{3}+*x*+1/2*x*=0, which has no Lie point symmetries but has a λ-symmetry vector field *t*/*x*^{2} [12]. For this λ-function, we can have the expression *I*/∂*x*=−*RS* and ∂*I*/∂*x*^{(1)}=−*R*, and solving it, we can obtain the null form *S* as *S*=−(*t*+*xx*^{(1)})/*x*^{2}. The corresponding integrating factor *R* can be found as *R*=1/*x*((*t*+2*xx*^{(1)})^{2}/4*x*^{4}+1). So we can find the rest of the integrability quantifiers. But in the case where *ξ*≠0, it is very difficult to solve the expression *ϕ*_{x(1)}. So the interconnections will work for arbitrary λ-symmetries, Jacobi last multipliers or Darboux polynomials, provided we are able to solve the corresponding equations.

## Authors' contributions

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

## Competing interests

We have no competing interests.

## Funding

R.M. acknowledges the University Grants Commission (UGC-RFSMS), Government of India, for providing a Research Fellowship. The work of V.K.C. is supported by an INSA young scientist project. 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 and is also supported by a DAE Raja Ramanna Fellowship and a DST Ramanna Fellowship programme.

- Received December 11, 2015.
- Accepted May 23, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.