On the Fiber Preserving Transformations for the Fifth-Order Ordinary Differential Equations

This paper is devoted to the study of the linearization problem of fifth-order ordinary differential equations by means of fiber preserving transformations.The necessary and sufficient conditions for linearization are obtained.The procedure for obtaining the linearizing transformations is provided in explicit form. Examples demonstrating the procedure of using the linearization theorems are presented.


Introduction
1.1.The Research Problem and Its Significance.In mathematics, a nonlinear equation is an equation which is not linear; that is, an equation which does not satisfy the superposition principle, or whose output is not directly proportional to its input.Less technically, a nonlinear equation is any problem where the variables to be solved for cannot be written as a linear combination of independent components.
Nonlinear problems are of interest to engineers, physicists, and mathematicians because most physical systems are inherently nonlinear in nature.Nonlinear equations are difficult to solve and give rise to interesting phenomena.While solving problems related to nonlinear ordinary differential equations, it is often expedient to simplify equations by a suitable change of variables.One of the fundamental methods to solve this relies upon the transformation of a given equation to another equation of standard form.The transformation may be to an equation of equal order or of greater or lesser order.In particular, the possibility that a given equation could be linearized, that is, transformed to a linear equation, was a most attractive proposition due to the special properties of linear differential equations.The reduction of an ordinary differential equation to a linear ordinary differential equation besides simplification allows us to construct an exact solution of the original equation.
One type of the classification problem is the equivalence problem.Two equations of differential equations are said to be equivalent if there exists an invertible transformation which transforms any solution of one equation to a solution of the other equation and vice versa.The linearization problem is a particular case of the equivalence problem, where one of the equations is a linear equation.It is one of the essential parts in the study of nonlinear equations.
The main difficulty in solving the linearization problem comes from the large number of complicated calculations.Because of this difficulty, no one attempts to solve this problem for nonlinear equations are higher than fourth.However if we can solve the linearization problem of fifthorder ordinary differential equations, then we should set a new process to solve the problems in Physics or Engineering.

Historical Review.
The linearization, that is, mapping a nonlinear differential equation into a linear differential equation, is an important tool in the theory of differential equations.The problem of linearization of ordinary differential equations attracted attention of mathematicians such as Lie and Cartan.The first linearization problem for ordinary differential equations was solved by Lie [1,2].He found the general form of all ordinary differential equations of secondorder that can be reduced to a linear equation by changing the independent and dependent variables.He showed that 2 Journal of Applied Mathematics any linearizable second-order equation should be at most cubic in the first-order derivative and provided a linearization test in terms of its coefficients.The linearization criterion is written through relative invariants of the equivalence group.Liouville [3] and Tresse [4] treated the equivalence problem for second-order ordinary differential equations in terms of relative invariants of the equivalence group of point transformations.There are other approaches for solving the linearization problem of a second-order ordinary differential equation.For example, one was developed by Cartan [5].The idea of his approach was to associate with every differential equation a uniquely defined geometric structure of a certain form.
In 1993, Bocharov et al. [6] considered the linearization problem of third-order with respect to point transformations.Grebot [7] studied the linearization of third-order ordinary differential equations by means of a restricted class of point transformations, namely,  = () and  = (, ).However, the problem was not completely solved.Complete criteria for linearization by means of point transformations were obtained by Ibragimov and Meleshko [8].
In 2008, Ibragimov et al. [9] solved the linearization problem for fourth-order ordinary differential equations by using point transformation.
Nowadays, the linearization problem of fifth-order ordinary differential equations via point transformations still is an unsolved one.

The Mapping of a Function by a Point Transformation
where  and  are sufficiently smooth functions, is called a point transformation.If   = 0, a transformation (1) is called a fiber preserving transformation.
Let us explain how a point transformation maps one function into another.
Assume that  0 () is a given function, the transformed function  0 () is defined by the following two steps.On the first step one has to solve with respect to  the equation  =  (,  0 ()) . ( Using the Inverse Function Theorem we find that  = () is a solution of this equation.The transformed function is determined by the formula Conversely, if one has the function  0 (), then for finding the function  0 () one has to solve the ordinary differential equation

Necessary Conditions of Linearization
We begin with investigating the necessary conditions for linearization.Recall that according to the Laguerre theorem a linear fifth-order ordinary differential equation has the form  (5) +  ()   +  ()   +  ()  = 0. ( Here we consider the fifth-order ordinary differential equations  (5) =  (, ,   ,   ,  (4) ) , which can be transformed to the linear equation ( 5) with  =  =  = 0 under the fiber preserving transformation So we arrive at the following theorem.

Formulation of the Linearization Theorem
We have shown in the previous section that every linearizable fifth-order ordinary differential equation belongs to the class of (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.