Linearizability of Nonlinear Third-Order Ordinary Differential Equations by Using a Generalized Linearizing Transformation

Abstract

We discuss the linearization problem of third-order ordinary differential equation under the generalized linearizing transformation. We identify the form of the linearizable equations and the conditions which allow the third-order ordinary differential equation to be transformed into the simplest linear equation. We also illustrate how to construct the generalized linearizing transformation. Some examples of linearizable equation are provided to demonstrate our procedure.

1. Introduction

There has been major interest in the nonlinear problems, since most equations are inherently nonlinear in nature. In general, the nonlinear problems are very difficult to solve explicitly. It is of interest to provide general criteria for the linearizability of nonlinear ordinary differential equations, as they can then be reduced to easily solvable equations. Therefore, the approach of investigating nonlinear ordinary differential equations via transforming to simpler ordinary differential equations becomes important and has been quite plentiful in analysis of physical problems. This includes the classical linearization problem of finding transformations that linearize a given ordinary differential equation. The linearization problem has been studied in many aspects. A short review can be found in [1, 2]. The tools commonly used for solving the linearization problem are the transformations such as point transformation, contact transformation, reduction of order, differential substitution, and generalized Sundman transformation. For this paper, we employ the extension of the generalized Sundman transformations.

The linearization problem for a second-order ordinary differential equation was investigated with respect to a generalized Sundman transformation by Duarte et al. [3] earlier. They obtained the form of the linearizable equations and the conditions which allow the second-order ordinary differential equation to be transformed into the free particle equation. A characterization of these equations that can be linearized by means of generalized Sundman transformations in terms of first integral and procedure for construction of linearizing transformations has been given by Muriel and Romero [4]. In [5], Mustafa et al. gave a new characterization of linearizable equations in terms of the coefficients of ordinary differential equations and one auxiliary function. In [6], Nakpim and Meleshko pointed out that the solution of the linearization problem for a second-order ordinary differential equation via the generalized Sundman transformation considered earlier by Duarte et al. [3] using the Laguerre form is not complete.

The linearization problem for a third-order ordinary differential equation was also investigated with respect to a generalized Sundman transformation [7, 8]. Criteria for a third-order ordinary differential equation to be equivalent to the linear equation with respect to a Sundman transformation were presented in [8]. The generalized Sundman transformation was also applied for obtaining necessary and sufficient conditions for a third-order ordinary differential equation to be equivalent to a linear equation in the Laguerre form [6]. Some applications of the generalized Sundman transformation to ordinary differential equations were considered in [9] and earlier papers, summarized in the book [10].

The linearization problem of a fourth-order ordinary differential equation with respect to generalized Sundman transformations was studied in [11]. They found the necessary and sufficient conditions which allow the fourth-order ordinary differential equation to be transformed into the simplest linear equation.

In this work, we expose a more general transformation, that is, the extension of the generalized Sundman transformation This transformation was studied in [12–14] where they designated the transformation as the generalized linearizing transformation. They showed that this transformation can be utilized to linearize a wider class of nonlinear ordinary differential equations and, in particular, certain equations which cannot be linearized by the nonpoint and invertible point transformations. If the function in (2) is independent of the variable , then it becomes a nonpoint transformation (vide (1)). On the other hand, if is a differentiable function, then it becomes an invertible point transformation. So (2) is a unified transformation as it includes nonpoint and invertible point transformations as special cases. An example of an equation which can be linearized by a transformation of the form (2) is given in [13]. It is worth noting that any second-order equation can be transformed by a transformation (2) into the free particle equation and that this is not so for third-order ordinary differential equations. Hence, the linearization problem using generalized linearizing transformations becomes interesting for ordinary differential equations of order greater than 2. In [12], the authors applied a particular class of transformations (2), where the function is linear with respect to .

We are now paying attention to the case where is a polynomial function in and in particular where it is linear in with coefficients which are arbitrary functions of and . To be specific, we focus here on the case Notice that for the case , the generalized linearizing transformation becomes a generalized Sundman transformation, so that we assume .

The paper is organized as follows. In Section 2, the necessary conditions of linearization of a third-order ordinary differential equation are presented. In Section 3, we get the theorems that yield criteria for a third-order ordinary differential equation to be linearizable via generalized linearizing transformations. Examples which illustrate the procedure of using the linearization theorems are presented in Section 4.

2. Necessary Conditions of Linearization

Here we consider a nonlinear third-order ordinary differential equation Our goal in this section is to describe all equations (4) which are equivalent with respect to generalized linearizing transformations to a linear equation We begin with investigating the necessary conditions for linearization, that is, the general form of third-order equation (4) that can be obtained from a linear equation (6) by any generalized linearizing transformation (5).

Applying a generalized linearizing transformation (5), one obtains the following transformation of the third-order derivatives: where is a total of derivatives. Substituting the resulting expression in linear equation (6) and setting , we arrive at the following equation: where and are functions of and determined as follows: Thus, we proved the theorem.

Theorem 1. Any third-order ordinary differential equation (4) obtained from a linear equation (6) by a generalized linearizing transformation (5) has to be in the form (8).

3. Formulation of the Linearization Theorem

We have shown in the previous section that every linearizable third-order ordinary differential equation belongs to the class of equations (8). In this section, we formulate the main theorems containing necessary and sufficient conditions for linearization as well as the methods for constructing the linearizing transformations.

For obtaining sufficient conditions, one has to solve the compatibility problem. Consider the representations of the coefficients and through the unknown functions and . According to our notation , we define the derivative as From (9), one can find the derivatives From (10), one obtains the condition Equation (11) defines the derivative So that equation (12) becomes The compatibility analysis depends on the value of . A complete study of all cases is given here.

3.1. Case

In this case, the forms of derivatives , and become Substituting into , one arrives at the condition Comparing the mixed derivatives , one gets the derivative In this case, is satisfied. Equations (12) and (13) give the conditions Comparing the mixed derivative , one obtains the condition Equation (14) provides the derivative where The relation gives the condition Comparing the mixed derivative , one arrives at the condition Solving (15), one finds the conditions

3.2. Case

From (20) and (13), one obtains the derivatives Comparing the mixed derivative , one obtains Equation (14) becomes where Further analysis of the compatibility depends on value of in (35): it is separated into two cases; that is, and .

3.2.1. Case

From (35), one finds Since this case , then too. Comparing the mixed derivatives , one gets the derivative where Substituting into , into and , one arrives at the conditions Equation (15) provides the conditions Comparing the mixed derivatives , , and , one gets the conditions

3.2.2. Case

From (35), one finds the condition Equation (15) gives the conditions From the mixed derivative , one finds the condition The relation becomes where The relation provides the condition Further study depends on . (i) Case

From (46), one gets the derivative Differentiating with respect to , one obtains the derivative where The relations provide the conditions  (ii) Case

From (46), one gets the condition Comparing the mixed derivative , one arrives at the condition All obtained results can be summarized in the following theorems.

Theorem 2. Sufficient conditions for (8) to be linearizable via the generalized linearizing transformation (5) with are equations (18), (22), (24), (25), (28), (29), (30), and (31).

Corollary 3. Provided that the sufficient conditions in Theorem 2 are satisfied, the transformation (5) mapping equation (8) to a linear equation (6) is obtained by solving the compatible system of equations (21), (23), and (26) for the functions , and .

Theorem 4. Sufficient conditions for equation (8) to be linearizable via the generalized linearizing transformation (5) with are as follows. (a)If , then the conditions are (18), (40), (41), and (42).(b)If , then the conditions are (18), (43), (44), (48), and (52).(c)If , then the conditions are (18), (43), (44), (53), and (54).

Corollary 5. Provided that the sufficient conditions in Theorem 4 are satisfied, the transformation (5) mapping equation (8) to a linear equation (6) is obtained by solving the following compatible system of equations for the functions , and : (a)(16), (17), (34), (37), and (38);(b)(16), (17), (34), (49), and (50);(c)(16), (17), (19), (32), (33), and (34).

4. Examples

For understanding the procedure of using the linearization theorems, we consider the following examples.

Example 1. Consider the nonlinear third-order ordinary differential equation It is an equation of the form (8) in Theorem 1 with the coefficients One can check that these coefficients obey the conditions in Theorem 2. Thus, (55) is linearizable via a generalized linearizing transformation. For finding the functions , and , we have to solve equations in Corollary 3, which become From the first equation of system (58), we get , and choosing , we have and this solution satisfies the second equation. Since , then we obtain System (59) becomes and one can take the simplest solution System (90) becomes so that we get the particular solution Thus, one obtains the linearizing transformation Hence, (55) is mapped by the transformation of (66) into the linear equation (6).

Example 2. Consider the nonlinear third-order ordinary differential equation It is an equation of the form (8) in Theorem 1 with the coefficients One can check that these coefficients obey the conditions in Theorem 4(a). Thus, (67) is linearizable via a generalized linearizing transformation. For finding the functions , and , we have to solve equations in Corollary 5(a), which become From the first equation of system (70), one can take the particular solution and this solution satisfies the second equation. Since , then we obtain System (71) becomes and one can take the simplest solution System (69) becomes so that we get the particular solution Thus, one obtains the linearizing transformation Hence, (67) is mapped by the transformation of (78) into the linear equation (6).

Example 3. Consider the nonlinear third-order ordinary differential equation Note that this equation can be reduced to an autonomous equation by the substitution and then to the second-order ordinary differential equation where . However, the latter equation is not linearizable by point transformations.

Equation (79) is an equation of the form (8) in Theorem 1 with the coefficients One can check that these coefficients obey the conditions in Theorem 4(c). Thus, (79) is linearizable via a generalized linearizing transformation. For finding the functions , and , we have to solve equations in Corollary 5(c), which become From the first equation of system (84), one can take the particular solution and this solution satisfies the second and third equations. Since , then we obtain System (85) becomes and one can take the particular solution System (83) becomes so that one obtains the particular solution of the first equation as and this solution satisfies the second equation. Then we get the linearizing transformation Hence, equation (79) is mapped by the transformation of (92) into the linear equation (6).

5. Conclusion

This paper is devoted to find the conditions which allow the third-order ordinary differential equation to be transformed into the simplest linear equation. Necessary conditions which guarantee that the third-order ordinary differential equation can be linearized are found in Theorem 1. Theorems 2 and 4 are sufficient conditions for the linearization problem. The linearizing transformation can be found by solving the compatible system in Corollaries 3 and 5. Finally, some examples are provided to demonstrate our procedure.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was financially supported by the National Research Council of Thailand under Grant no. R2557B057.