Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011, Article IDΒ 452689, 21 pages
http://dx.doi.org/10.5402/2011/452689
Research Article

Linearization of Two Second-Order Ordinary Differential Equations via Fiber Preserving Point Transformations

School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand

Received 1 July 2011; Accepted 4 August 2011

Academic Editor: M.Β Escobedo

Copyright Β© 2011 Sakka Sookmee and Sergey V. Meleshko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The necessary form of a linearizable system of two second-order ordinary differential equations 𝑦1ξ…žξ…ž=𝑓1(π‘₯,𝑦1,𝑦2,π‘¦ξ…ž1,π‘¦ξ…ž2), 𝑦2ξ…žξ…ž=𝑓2(π‘₯,𝑦1,𝑦2,π‘¦ξ…ž1,π‘¦ξ…ž2) is obtained. Some other necessary conditions were also found. The main problem studied in the paper is to obtain criteria for a system to be equivalent to a linear system with constant coefficients under fiber preserving transformations. A linear system with constant coefficients is chosen because of its simplicity in finding the general solution. Examples demonstrating the procedure of using the linearization theorems are presented.

1. Introduction

Almost all physical applications of differential equations are based on nonlinear equations, which in general are very difficult to solve explicitly. Ordinary differential equations play a significant role in the theory of differential equations. In the 19th century, one of the most important problems in analysis was the problem of classification of ordinary differential equations [1–4].

One type of the classification problem is the equivalence problem. Two systems of differential equations are said to be equivalent if there exists an invertible transformation which transforms any solution of one system to a solution of the other system. The linearization problem is a particular case of the equivalence problem, where one of the systems is a linear system. It is one of essential parts in the studies of nonlinear equations.

1.1. Linearization Problem

The analysis of the linearization problem for a single ordinary differential equation was started by Lie [1]. He gave criteria for linearization of a second-order ordinary differential equation by an invertible change of the independent and dependent variables (point transformation). Cartan [4] developed another approach for solving the linearization problem of a second-order ordinary differential equation.

Later the linearization problem was also considered with respect to other types of transformations, for example, contact and generalized Sundman transformations. These transformations have been applied to third-order and fourth-order ordinary differential equations, see [5–16] and references therein. It is worth to note that fiber preserving transformations, where the change of variables only depends on the independent variable, have played a special role: either only such transformations were studied [6], or one needs to study them separately during compatibility analysis [10, 11, 13, 17].

The linearization problem for a system of second-order ordinary differential equations was studied in [18–24]. In [18], necessary and sufficient conditions for a system of 𝑛β‰₯2 second-order ordinary differential equations to be equivalent to the free particle equations were given. Particular class of systems of two (𝑛=2) second-order ordinary differential equations were considered in [20]. In [19], criteria for linearization of a system of two second-order ordinary differential equations are related with the existence of an admitted four-dimensional Lie algebra. Some first-order and second-order relative invariants with respect to point transformations for a system of two ordinary differential equations were obtained in [21]. A new method of linearizing a system of equations is proposed in [22], where a given system of equations is reduced to a single equation to which the Lie theorem on linearization is applied.

1.2. Canonical Forms of a Linear System of Two Second-Order ODEs

The general form of a normal linear system of 𝑛 second-order ordinary differential equations isΜˆΜ‡π‘£+𝐢𝑣+𝐷𝑣+𝐸=0,(1.1) where 𝑣=𝑣(𝑑) and 𝐸=𝐸(𝑑) are vectors, 𝐢=𝐢(𝑑) and 𝐷=𝐷(𝑑) are 𝑛×𝑛 square matrices. It can be shown [19] that there exists a change 𝑒=π‘ˆπ‘£ such that system (1.1) is reduced to one of the following forms: either Μˆπ‘’+𝐾1̇𝑒=0(1.2) orΜˆπ‘’+𝐾𝑒=0.(1.3) Here π‘ˆ(𝑑), 𝐾=𝐾(𝑑), and 𝐾1=𝐾1(𝑑) are 𝑛×𝑛 square matrices. Thus the problem of linearization via point transformations consists of solving the problem of reducibility of a system of second-order ordinary differential equations to one of these forms. In the present paper, the second canonical form (1.3) is used.

It is also worth to note that system (1.3) can be further simplified to one of three forms:βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πΎ=π‘Ž00𝑏,𝐾=π‘Ž10π‘Ž,𝐾=π‘Žπ‘βˆ’π‘π‘Ž,(1.4) where π‘Ž=π‘Ž(𝑑), 𝑏=𝑏(𝑑), and 𝑐=𝑐(𝑑)β‰ 0 are real valued (not complex) functions.

One of the main motivations for studying the linearization problem is the possibility of finding the general solution. Notice that even after finding the linearizing transformation one has to solve a linear system of second-order ordinary differential equations. The simplest case is where 𝐾=0. More general and also not complicated is the case where the matrix 𝐾 is constant. For example, for 𝑛=2, this case leads to solving either a simple linear fourth-order ordinary differential equation with constant coefficients or two simple linear second-order ordinary differential equations. Indeed, for 𝑛=2, system (1.3) isΜˆπ‘’1+π‘˜1𝑒1+π‘˜3𝑒2=0,Μˆπ‘’2+π‘˜4𝑒1+π‘˜2𝑒2=0,(1.5) where π‘˜π‘–, (𝑖=1,2,3,4) are constant. If π‘˜3β‰ 0, then finding 𝑒2 from the first equation of (1.5) and substituting it into the second equation of (1.5), one obtains a fourth-order ordinary differential equation𝑒1(4)+ξ€·π‘˜1+π‘˜2ξ€ΈΜˆπ‘’1+ξ€·π‘˜1π‘˜2βˆ’π‘˜3π‘˜4𝑒1=0.(1.6) Here,βŽ›βŽœβŽœβŽπ‘˜πΎ=1π‘˜3π‘˜4π‘˜2⎞⎟⎟⎠.(1.7) The general solution of the last equation depends on the roots πœ† of the characteristic equationπœ†4+ξ€·π‘˜1+π‘˜2ξ€Έπœ†2+ξ€·π‘˜1π‘˜2βˆ’π‘˜3π‘˜4ξ€Έ=0.(1.8) The solution is similar for π‘˜4β‰ 0. On the other hand, if π‘˜3=0 and π‘˜4=0, then system (1.1) is decoupled:Μˆπ‘’1+π‘˜1𝑒1=0,Μˆπ‘’2+π‘˜2𝑒2=0.(1.9) Notice also that if in this case π‘˜1=π‘˜2, then the last system of equations is equivalent to the system of two trivial equations π‘§ξ…žξ…ž=0.

1.3. The Problem Solved in the Present Paper

The present paper deals with a system of two second-order ordinary differential equations𝑦1ξ…žξ…ž=𝐹1ξ€·π‘₯,𝑦1,𝑦2,π‘¦ξ…ž1,π‘¦ξ…ž2ξ€Έ,𝑦2ξ…žξ…ž=𝐹2ξ€·π‘₯,𝑦1,𝑦2,π‘¦ξ…ž1,π‘¦ξ…ž2ξ€Έ.(1.10) In the next section, the form of a linearizable system (1.10) is obtained. This form coincides with that obtained in [18]. Some invariants of this form with respect to the general set of point transformations𝑑=πœ‘π‘₯,𝑦1,𝑦2ξ€Έ,𝑒1=πœ“1ξ€·π‘₯,𝑦1,𝑦2ξ€Έ,𝑒2=πœ“2ξ€·π‘₯,𝑦1,𝑦2ξ€Έ(1.11) related with a linearizable systems (1.10) were obtained in [21]. The main result of the paper consists of necessary and sufficient conditions for system (1.10) to be equivalent with respect to a fiber preserving point transformation𝑑=πœ‘(π‘₯),𝑒1=πœ“1ξ€·π‘₯,𝑦1,𝑦2ξ€Έ,𝑒2=πœ“2ξ€·π‘₯,𝑦1,𝑦2ξ€Έ(1.12) to system (1.5).

The paper is organized as follows. In Section 2, the necessary form of a linearizable system of two second-order ordinary differential equations is presented. In Section 3, the main results of the paper are given: necessary and sufficient criteria for a system of two second-order ordinary differential equations to be equivalent via fiber preserving point transformations (1.12) to a system of the form (1.5). During the study presented in Section 3, we also obtained some necessary conditions for linearizability for the general case of point transformations (1.11) and for the general case of the matrix 𝐾(𝑑). These conditions are shown in Section 4. Examples demonstrating the procedure of using the linearization theorems are presented in Section 5.

2. Necessary Form of a Linearizable System (1.10)

For obtaining necessary conditions for system (1.10) to be linearizable via point transformations (1.11), one assumes that system (1.10) is obtained from the linear system of differential equations (1.3) by an invertible transformation (1.11). The derivatives are changed by the formulae π‘’ξ…ž1=𝑔1ξ€·π‘₯,𝑦1,𝑦2,π‘¦ξ…ž1,π‘¦ξ…ž2ξ€Έ=𝐷π‘₯πœ“1𝐷π‘₯πœ‘,𝑒1ξ…žξ…ž=𝐷π‘₯𝑔1𝐷π‘₯πœ‘,π‘’ξ…ž2=𝑔2ξ€·π‘₯,𝑦1,𝑦2,π‘¦ξ…ž1,π‘¦ξ…ž2ξ€Έ=𝐷π‘₯πœ“2𝐷π‘₯πœ‘,𝑒2ξ…žξ…ž=𝐷π‘₯𝑔2𝐷π‘₯πœ‘,(2.1) where𝐷π‘₯=πœ•πœ•π‘₯+π‘¦ξ…ž1πœ•πœ•π‘¦1+π‘¦ξ…ž2πœ•πœ•π‘¦2+𝑦1ξ…žξ…žπœ•πœ•π‘¦ξ…ž1+𝑦2ξ…žξ…žπœ•πœ•π‘¦ξ…ž2.(2.2) Replacing π‘’ξ…ž1, 𝑒1ξ…žξ…ž, π‘’ξ…ž2, and 𝑒2ξ…žξ…ž in system (1.3), it becomes𝑦1ξ…žξ…ž=π‘¦ξ…ž1ξ‚€π‘Ž11π‘¦ξ…ž12+π‘Ž12π‘¦ξ…ž1π‘¦ξ…ž2+π‘Ž13π‘¦ξ…ž22+π‘Ž14π‘¦ξ…ž12+π‘Ž15π‘¦ξ…ž1π‘¦ξ…ž2+π‘Ž16π‘¦ξ…ž22+π‘Ž17π‘¦ξ…ž1+π‘Ž18π‘¦ξ…ž2+π‘Ž19,𝑦2ξ…žξ…ž=π‘¦ξ…ž2ξ‚€π‘Ž11π‘¦ξ…ž12+π‘Ž12π‘¦ξ…ž1π‘¦ξ…ž2+π‘Ž13π‘¦ξ…ž22+π‘Ž24π‘¦ξ…ž12+π‘Ž25π‘¦ξ…ž1π‘¦ξ…ž2+π‘Ž26π‘¦ξ…ž22+π‘Ž27π‘¦ξ…ž1+π‘Ž28π‘¦ξ…ž2+π‘Ž29,(2.3) where the coefficients π‘Žπ‘–π‘— are expressed through the functions πœ‘, πœ“1, πœ“2, their partial derivatives, and the entries of the matrix 𝐾=(π‘˜π‘–π‘—(𝑑)) as follows:π‘Ž11=ξ‚€β„Ž1πœ“1𝑦1𝑦1+β„Ž2πœ“2𝑦1𝑦1βˆ’π‘£πœ‘π‘¦1𝑦1+𝑓1πœ‘π‘¦13βˆ’π‘“2πœ‘π‘¦12πœ‘π‘¦2Δ,π‘Ž12=2ξ‚€β„Ž1πœ“1𝑦1𝑦2+β„Ž2πœ“2𝑦1𝑦2βˆ’π‘£πœ‘π‘¦1𝑦2+𝑓1πœ‘π‘¦12πœ‘π‘¦2βˆ’π‘“2πœ‘π‘¦1πœ‘π‘¦22Δ,π‘Ž13=ξ‚€β„Ž1πœ“1𝑦2𝑦2+β„Ž2πœ“2𝑦2𝑦2βˆ’π‘£πœ‘π‘¦2𝑦2+𝑓1πœ‘π‘¦1πœ‘π‘¦22βˆ’π‘“2πœ‘π‘¦23Δ,π‘Ž14=ξ‚€2β„Ž1πœ“1π‘₯𝑦1+2β„Ž2πœ“2π‘₯𝑦1+β„Ž3πœ“1𝑦1𝑦1+β„Ž4πœ“2𝑦1𝑦1+β„Ž5πœ‘π‘¦1𝑦1βˆ’2π‘£πœ‘π‘₯𝑦1+3𝑓1πœ‘π‘₯πœ‘π‘¦12βˆ’2𝑓2πœ‘π‘₯πœ‘π‘¦1πœ‘π‘¦2ξ‚βˆ’π”Ξ”,π‘Ž15=2ξ‚€β„Ž1πœ“1π‘₯𝑦2+β„Ž2πœ“2π‘₯𝑦2+β„Ž3πœ“1𝑦1𝑦2+β„Ž4πœ“2𝑦1𝑦2+β„Ž5πœ‘π‘¦1𝑦2βˆ’2π‘£πœ‘π‘₯𝑦2+2𝑓1πœ‘π‘₯πœ‘π‘¦1πœ‘π‘¦2βˆ’π‘“2πœ‘π‘₯πœ‘π‘¦22ξ‚βˆ’β„œΞ”,π‘Ž16=ξ‚€β„Ž3πœ“1𝑦2𝑦2+β„Ž4πœ“2𝑦2𝑦2+β„Ž5πœ‘π‘¦2𝑦2+𝑓1πœ‘π‘₯πœ‘π‘¦22βˆ’π‘“3πœ‘π‘¦23Δ,π‘Ž17=ξ‚€β„Ž1πœ“1π‘₯π‘₯+β„Ž2πœ“2π‘₯π‘₯+2β„Ž3πœ“1π‘₯𝑦1+2β„Ž4πœ“2π‘₯𝑦1+2β„Ž5πœ‘π‘₯𝑦1βˆ’π‘£πœ‘π‘₯π‘₯+3𝑓1πœ‘π‘₯2πœ‘π‘¦1βˆ’π‘“2πœ‘π‘₯2πœ‘π‘¦2βˆ’2𝑓3πœ‘π‘₯πœ‘π‘¦1Δ,π‘Ž18=2ξ‚€β„Ž3πœ“1π‘₯𝑦2+β„Ž4πœ“2π‘₯𝑦2+2β„Ž5πœ‘π‘₯𝑦2+𝑓1πœ‘π‘₯2πœ‘π‘¦2βˆ’π‘“3πœ‘π‘₯πœ‘π‘¦22Δ,π‘Ž19=ξ€·β„Ž3πœ“1π‘₯π‘₯+β„Ž4πœ“2π‘₯π‘₯+β„Ž5πœ‘π‘₯π‘₯+𝑓1πœ‘π‘₯3βˆ’π‘“3πœ‘π‘₯2πœ‘π‘¦2ξ€ΈΞ”,π‘Ž24=ξ‚€β„Ž6πœ“1𝑦1𝑦1+β„Ž7πœ“2𝑦1𝑦1+β„Ž8πœ‘π‘¦1𝑦1βˆ’π‘“2πœ‘π‘₯πœ‘π‘¦12+𝑓3πœ‘π‘¦13Δ,π‘Ž25=2ξ‚€β„Ž1πœ“1π‘₯𝑦1+β„Ž2πœ“2π‘₯𝑦1βˆ’π‘£πœ‘π‘₯𝑦1+β„Ž6πœ“1𝑦1𝑦2+β„Ž7πœ“2𝑦1𝑦2+𝑓1πœ‘π‘₯πœ‘π‘₯πœ‘π‘¦12βˆ’2𝑓2πœ‘π‘₯πœ‘π‘¦1πœ‘π‘¦2βˆ’π‘“3πœ‘π‘¦12πœ‘π‘¦2Δ,π‘Ž26=ξ‚€2β„Ž1πœ“1π‘₯𝑦2+2β„Ž2πœ“2π‘₯𝑦2βˆ’2π‘£πœ‘π‘₯𝑦2+β„Ž6πœ“1𝑦2𝑦2+β„Ž7πœ“2𝑦2𝑦2+β„Ž8πœ‘π‘¦2𝑦2+2𝑓1πœ‘π‘₯πœ‘π‘¦1πœ‘π‘¦2βˆ’3𝑓2πœ‘π‘₯πœ‘π‘¦22ξ‚βˆ’β„œΞ”,π‘Ž27=2ξ‚€β„Ž6πœ“1π‘₯𝑦1+β„Ž7πœ“2π‘₯𝑦1+β„Ž8πœ‘π‘₯𝑦1βˆ’π‘“2πœ‘π‘₯2πœ‘π‘¦1+𝑓3πœ‘π‘₯πœ‘π‘¦12Δ,π‘Ž28=ξ‚€β„Ž1πœ“1π‘₯π‘₯+β„Ž2πœ“2π‘₯π‘₯βˆ’π‘£πœ‘π‘₯π‘₯+2β„Ž6πœ“1π‘₯𝑦2+2β„Ž7πœ“2π‘₯𝑦2+2β„Ž8πœ‘π‘₯𝑦2+𝑓1πœ‘π‘₯2πœ‘π‘¦1βˆ’3𝑓2πœ‘π‘₯2πœ‘π‘¦2+π”˜Ξ”,π‘Ž29=ξ€·β„Ž6πœ“1π‘₯π‘₯+β„Ž7πœ“2π‘₯π‘₯+β„Ž8πœ‘π‘₯π‘₯βˆ’π‘“2πœ‘π‘₯3+𝑓3πœ‘π‘₯2πœ‘π‘¦1ξ€ΈΞ”,(2.4) where 𝔍 denotes 𝑓3πœ‘π‘¦12πœ‘π‘¦2, β„œ denotes 𝑓3πœ‘π‘¦1πœ‘π‘¦22, π”˜ denotes 2𝑓3πœ‘π‘₯πœ‘π‘¦1πœ‘π‘¦2, and Ξ”β‰ 0 is the Jacobian of the change of variables  (1.11)ξ‚€πœ‘Ξ”=π‘₯πœ“1𝑦1πœ“2𝑦2βˆ’πœ‘π‘₯πœ“1𝑦2πœ“2𝑦1βˆ’πœ‘π‘¦1πœ“1π‘₯πœ“2𝑦2+πœ‘π‘¦1πœ“1𝑦2πœ“2π‘₯+πœ‘π‘¦2πœ“1π‘₯πœ“2𝑦1βˆ’πœ‘π‘¦2πœ“1𝑦1πœ“2π‘₯,𝑓1=πœ“1𝑦2ξ€·π‘˜22πœ“2+π‘˜21πœ“1ξ€Έβˆ’πœ“2𝑦2ξ€·π‘˜11πœ“1+π‘˜12πœ“2ξ€Έ,𝑓2=πœ“1𝑦1ξ€·π‘˜22πœ“2+π‘˜21πœ“1ξ€Έβˆ’πœ“2𝑦1ξ€·π‘˜11πœ“1+π‘˜12πœ“2ξ€Έ,𝑓3=πœ“1π‘₯ξ€·π‘˜22πœ“2+π‘˜21πœ“1ξ€Έβˆ’πœ“2π‘₯ξ€·π‘˜11πœ“1+π‘˜12πœ“2ξ€Έ,𝑣=πœ“1𝑦2πœ“2𝑦1βˆ’πœ“1𝑦1πœ“2𝑦2,β„Ž1=πœ‘π‘¦2πœ“2𝑦1βˆ’πœ‘π‘¦1πœ“2𝑦2,β„Ž2=πœ‘π‘¦1πœ“1𝑦2βˆ’πœ‘π‘¦2πœ“1𝑦1,β„Ž3=πœ‘π‘¦2πœ“2π‘₯βˆ’πœ‘π‘₯πœ“2𝑦2,β„Ž4=πœ‘π‘₯πœ“1𝑦2βˆ’πœ‘π‘¦2πœ“1π‘₯,β„Ž5=πœ“1π‘₯πœ“2𝑦2βˆ’πœ“1𝑦2πœ“2π‘₯,β„Ž6=πœ‘π‘₯πœ“2𝑦1βˆ’πœ‘π‘¦1πœ“2π‘₯,β„Ž7=πœ‘π‘¦1πœ“1π‘₯βˆ’πœ‘π‘₯πœ“1𝑦1,β„Ž8=πœ“1𝑦1πœ“2π‘₯βˆ’πœ“1π‘₯πœ“2𝑦1.(2.5)

Equation (2.3) presents the necessary form of a system of two second-order ordinary differential equations which can be mapped into a linear equation (1.3) via point transformations.

3. Sufficient Conditions for Equivalency to (1.5) via Fiber Preserving Transformations

For obtaining sufficient conditions of linearizability of system (2.3), one has to solve the compatibility problem of the system of (2.4), considering it as an overdetermined system of partial differential equations for the functions πœ“1, πœ“2, and πœ‘ with given coefficients π‘Žπ‘–π‘— of system (2.3). The compatibility analysis depends on the value of πœ“1𝑦1.

The next part of the present paper deals with a fiber preserving set of point transformations (1.12):𝑑=πœ‘(π‘₯),𝑒1=πœ“1ξ€·π‘₯,𝑦1,𝑦2ξ€Έ,𝑒2=πœ“2ξ€·π‘₯,𝑦1,𝑦2ξ€Έ,(3.1) and constant matrixβŽ›βŽœβŽœβŽπ‘˜πΎ=1π‘˜3π‘˜4π‘˜2⎞⎟⎟⎠.(3.2)

3.1. Case πœ“1𝑦1β‰ 0

Substitution of πœ‘π‘¦1=0 and πœ‘π‘¦2=0 into (2.4) gives π‘Ž11=0,π‘Ž12=0,π‘Ž13𝑣=0,(3.3)π‘₯=ξ€·2πœ‘π‘₯π‘₯π‘£βˆ’πœ‘π‘₯π‘£ξ€·π‘Ž17+π‘Ž28ξ€Έξ€Έξ€·2πœ‘π‘₯ξ€Έ,𝑣𝑦1𝑣=βˆ’2π‘Ž14+π‘Ž25ξ€Έ2,𝑣𝑦2π‘£ξ€·π‘Ž=βˆ’15+2π‘Ž26ξ€Έ2,πœ“(3.4)1π‘₯π‘₯=ξ‚€πœ‘π‘₯π‘₯πœ“1π‘₯βˆ’πœ‘π‘₯3ξ€·π‘˜1πœ“1+π‘˜3πœ“2ξ€Έβˆ’πœ‘π‘₯πœ“1𝑦1π‘Ž19βˆ’πœ‘π‘₯πœ“1𝑦2π‘Ž29ξ‚πœ‘π‘₯,πœ“(3.5)1π‘₯𝑦1=ξ‚€πœ‘π‘₯π‘₯πœ“1𝑦1βˆ’πœ‘π‘₯πœ“1𝑦1π‘Ž17βˆ’πœ‘π‘₯πœ“1𝑦2π‘Ž272πœ‘π‘₯ξ€Έ,πœ“(3.6)1π‘₯𝑦2=ξ‚€πœ‘π‘₯π‘₯πœ“1𝑦2βˆ’πœ‘π‘₯πœ“1𝑦1π‘Ž18βˆ’πœ‘π‘₯πœ“1𝑦2π‘Ž282πœ‘π‘₯ξ€Έ,πœ“(3.7)1𝑦1𝑦1ξ‚€πœ“=βˆ’1𝑦1π‘Ž14+πœ“1𝑦2π‘Ž24ξ‚πœ“,(3.8)1𝑦1𝑦2ξ‚€πœ“=βˆ’1𝑦1π‘Ž15+πœ“1𝑦2π‘Ž252,πœ“(3.9)1𝑦2𝑦2ξ‚€πœ“=βˆ’1𝑦1π‘Ž16+πœ“1𝑦2π‘Ž26ξ‚πœ“,(3.10)2𝑦2=ξ‚€πœ“1𝑦2πœ“2𝑦1ξ‚βˆ’π‘£πœ“1𝑦1,πœ“(3.11)2𝑦1𝑦1=ξ‚€π‘Ž24π‘£βˆ’πœ“1𝑦1πœ“2𝑦1π‘Ž14βˆ’πœ“1𝑦2πœ“2𝑦1π‘Ž24ξ‚πœ“1𝑦1,πœ“(3.12)2π‘₯𝑦1=ξ‚€πœ‘π‘₯π‘₯πœ“1𝑦1πœ“2𝑦1βˆ’πœ‘π‘₯πœ“1𝑦1πœ“2𝑦1π‘Ž17βˆ’πœ‘π‘₯πœ“1𝑦2πœ“2𝑦1π‘Ž27+πœ‘π‘₯π‘Ž27𝑣2πœ‘π‘₯πœ“1𝑦1ξ‚πœ“,(3.13)2π‘₯π‘₯=ξ‚€πœ‘π‘₯π‘₯πœ“1𝑦1πœ“2π‘₯βˆ’πœ‘π‘₯3πœ“1𝑦1ξ€·π‘˜2πœ“2+π‘˜4πœ“1ξ€Έβˆ’πœ‘π‘₯πœ“1𝑦1πœ“2𝑦1π‘Ž19βˆ’πœ‘π‘₯πœ“1𝑦2πœ“2𝑦1π‘Ž29+πœ‘π‘₯π‘Ž29π‘£ξ‚ξ‚€πœ‘π‘₯πœ“1𝑦1,(3.14) where Ξ”=βˆ’πœ‘π‘₯𝑣≠0. Comparing the mixed derivatives (𝑣π‘₯)𝑦1=(𝑣𝑦1)π‘₯,(𝑣π‘₯)𝑦2=(𝑣𝑦2)π‘₯ and (𝑣𝑦2)𝑦1=(𝑣𝑦1)𝑦2, one obtains 2π‘Žπ‘₯14βˆ’π‘Žπ‘¦171+π‘Žπ‘₯25βˆ’π‘Žπ‘¦281=0,π‘Žπ‘₯15βˆ’π‘Žπ‘¦172+2π‘Žπ‘₯26βˆ’π‘Žπ‘¦282π‘Ž=0,𝑦151βˆ’2π‘Žπ‘¦142βˆ’π‘Žπ‘¦252+2π‘Žπ‘¦261=0.(3.15) Considering the conditions (πœ“π‘–π‘₯π‘₯)𝑦𝑗=(πœ“π‘–π‘₯𝑦𝑗)π‘₯(𝑖,𝑗=1,2), one has πœ‘π‘₯π‘₯π‘₯=ξ‚€3πœ‘2π‘₯π‘₯πœ“1𝑦1βˆ’4πœ‘π‘₯4πœ“1𝑦1π‘˜1βˆ’4πœ‘π‘₯4πœ“2𝑦1π‘˜3+πœ‘π‘₯2πœ“1𝑦1ξ€·πœ†20βˆ’πœ†16ξ€Έ+πœ‘π‘₯2πœ“1𝑦2πœ†122πœ‘π‘₯πœ“1𝑦1ξ‚π‘˜,(3.16)4=ξ‚€4πœ‘π‘₯2πœ“1𝑦1πœ“2𝑦1ξ€·π‘˜1βˆ’π‘˜2ξ€Έ+4πœ‘π‘₯2πœ“2𝑦12π‘˜3βˆ’π‘£πœ†124πœ‘π‘₯2πœ“1𝑦12ξ‚π‘˜,(3.17)3=ξ‚€πœ“1𝑦22πœ†12βˆ’πœ“1𝑦12πœ†15βˆ’πœ“1𝑦1πœ“1𝑦2πœ†164πœ‘π‘₯2𝑣,π‘˜2=ξ‚€4πœ‘π‘₯2πœ“1𝑦1π‘˜1π‘£βˆ’2πœ“1𝑦12πœ“2𝑦1πœ†15βˆ’2πœ“1𝑦1πœ“1𝑦2πœ“2𝑦1πœ†16+πœ“1𝑦1π‘£πœ†16+2πœ“1𝑦22πœ“2𝑦1πœ†12βˆ’2πœ“1𝑦2π‘£πœ†124πœ‘π‘₯2πœ“1𝑦1𝑣.(3.18) Here, the functions πœ†π‘›(π‘₯,𝑦1,𝑦2) are defined through π‘Žπ‘–π‘—(π‘₯,𝑦1,𝑦2) and their derivatives (presented in the appendix).

Equating the mixed derivatives (πœ“π‘–π‘₯𝑦1)𝑦2=(πœ“π‘–π‘₯𝑦2)𝑦1, (πœ“π‘–π‘₯𝑦1)𝑦1=(πœ“π‘–π‘¦1𝑦1)π‘₯, (πœ“1π‘₯𝑦2)𝑦𝑖=(πœ“1𝑦𝑖𝑦2)π‘₯, (πœ“1𝑦1𝑦𝑖)𝑦2=(πœ“1𝑦𝑖𝑦2)𝑦1, (πœ“2𝑦1𝑦1)𝑦2=(πœ“2𝑦2)𝑦1𝑦1 and using the conditions (π‘˜π‘–+1)𝑦2=0, (𝑖=1,2), one obtainsπœ†π‘›=0(𝑛=1,2,…,11).(3.19) Note that (πœ“1π‘₯𝑦2)𝑦1βˆ’(πœ“1𝑦1𝑦2)π‘₯=0 is satisfied. Differentiating (3.16) and (3.17) with respect to 𝑦1 and 𝑦2, one has2πœ†π‘¦151βˆ’2π‘Ž14πœ†15βˆ’π‘Ž15πœ†16+π‘Ž25πœ†15=0,πœ†27+𝑗=0(𝑗=0,1,2).(3.20)

Equations (π‘˜π‘–)π‘₯=0(𝑖=2,3,4), become 4πœ‘π‘₯π‘₯πœ†12+πœ‘π‘₯πœ†14=0,(3.21)2πœ‘π‘₯π‘₯πœ†16βˆ’πœ‘π‘₯ξ€·πœ†π‘₯16+π‘Ž18πœ†12βˆ’π‘Ž27πœ†15ξ€Έ=0,(3.22)4πœ‘π‘₯π‘₯πœ†15+πœ‘π‘₯ξ€·π‘Ž17πœ†15βˆ’2πœ†π‘₯15+π‘Ž18πœ†16βˆ’π‘Ž28πœ†15ξ€Έ=0.(3.23) Further analysis of the compatibility depends on the values of the coefficients πœ†12, πœ†16 and πœ†15 of the last three equations (3.21)–(3.23).

3.1.1. Case πœ†12β‰ 0

Substituting πœ‘π‘₯π‘₯, found from (3.21), into (3.23) and (3.22), one obtainsπ‘Ž17πœ†12πœ†15βˆ’2πœ†π‘₯15πœ†12+π‘Ž18πœ†12πœ†16βˆ’π‘Ž28πœ†12πœ†15βˆ’πœ†14πœ†15=0,2πœ†π‘₯16πœ†12+2π‘Ž18πœ†212βˆ’2π‘Ž27πœ†12πœ†15+πœ†14πœ†16=0.(3.24) Differentiating (3.21) with respect to 𝑦1 and 𝑦2, one hasπœ†π‘¦121πœ†14βˆ’πœ†π‘¦141πœ†12=0,πœ†π‘¦122πœ†14βˆ’πœ†π‘¦142πœ†12=0.(3.25) Equation (3.16) becomesπ‘˜1=ξ‚€16πœ“1𝑦12πœ“2𝑦1πœ†212πœ†15+16πœ“1𝑦1πœ“1𝑦2πœ“2𝑦1πœ†212πœ†16+16πœ“1𝑦1πœ†212π‘£ξ€·πœ†20βˆ’πœ†16ξ€Έ+4πœ“1𝑦1πœ†12𝑣2πœ†π‘₯14+π‘Ž17πœ†14βˆ’π‘Ž28πœ†14ξ€Έ+πœ“1𝑦1πœ†14𝑣4π‘Ž27πœ†16+5πœ†14ξ€Έβˆ’16πœ“1𝑦22πœ“2𝑦1πœ†312+16πœ“1𝑦2πœ†312𝑣/ξ‚€64πœ‘π‘₯2πœ“1𝑦1πœ†212𝑣.(3.26) Differentiating (3.26) with respect to π‘₯, one gets the condition32πœ†312πœ†17+8πœ†212πœ†18+2πœ†12πœ†19+πœ†14ξ€·8π‘Ž227πœ†216+18π‘Ž27πœ†14πœ†16+15πœ†214ξ€Έ=0.(3.27) Notice that (π‘˜1)𝑦1=0 and (π‘˜1)𝑦2=0 are satisfied. Hence, there are no new conditions for the functions πœ‘(π‘₯), πœ“1(π‘₯,𝑦1,𝑦2), and πœ“2(π‘₯,𝑦1,𝑦2). In summary, the criteria for linearization are conditions (3.3), (3.15), (3.19), (3.20), (3.24), (3.25), and (3.27).

3.1.2. Case πœ†12=0 and πœ†16β‰ 0

Since πœ†12=0 and πœ‘π‘₯β‰ 0, (3.21) leads to the condition πœ†14=0, and (3.22) becomesπœ‘π‘₯π‘₯=πœ‘π‘₯πœ†π‘₯16ξ€·2πœ†16ξ€Έ.(3.28) Substituting πœ‘π‘₯π‘₯ into (3.23) and (3.16), one gets2ξ€·πœ†π‘₯16πœ†15βˆ’πœ†π‘₯15πœ†16ξ€Έ+πœ†15πœ†16ξ€·π‘Ž17βˆ’π‘Ž28ξ€Έ+π‘Ž18πœ†216π‘˜=0,(3.29)1=ξ‚€4πœ“1𝑦1πœ“2𝑦1πœ†216πœ†15+4πœ“1𝑦2πœ“2𝑦1πœ†316βˆ’4πœ†316𝑣+4πœ†216πœ†23π‘£βˆ’4πœ†16πœ†16π‘₯π‘₯𝑣+5πœ†π‘₯216𝑣16πœ‘π‘₯2πœ†216𝑣.(3.30) Equation (π‘˜1)π‘₯=0 leads to the condition 8πœ†316πœ†21+4πœ†216πœ†22+18πœ†16πœ†16π‘₯π‘₯πœ†π‘₯16βˆ’15πœ†π‘₯316=0.(3.31) Note that (πœ‘π‘₯π‘₯)𝑦𝑖=0 and (π‘˜1)𝑦𝑖=0, (𝑖=1,2) are satisfied. Hence, there are no other conditions for the functions πœ‘(π‘₯), πœ“1(π‘₯,𝑦1,𝑦2), and πœ“2(π‘₯,𝑦1,𝑦2). Summarizing, the linearization criteria in the case πœ†12=0 and πœ†16β‰ 0 are conditions (3.3), (3.15), (3.19), (3.20), (3.29), and (3.31).

3.1.3. Case πœ†12=0,πœ†16=0, and πœ†15β‰ 0

Substituting πœ‘π‘₯π‘₯, found from (3.23), into (3.16), one hasπ‘˜1=ξ‚€16πœ“1𝑦1πœ“2𝑦1πœ†315+πœ†26𝑣64πœ‘π‘₯2πœ†215𝑣.(3.32) Differentiating (3.32) with respect to π‘₯, one getsπœ†315πœ†24+2πœ†215πœ†25βˆ’120πœ†π‘₯315+36πœ†15πœ†π‘₯15ξ€·4πœ†15π‘₯π‘₯+πœ†π‘₯15π‘Ž17βˆ’πœ†π‘₯15π‘Ž28ξ€Έ=0.(3.33) Note that the equations (πœ‘π‘₯π‘₯)𝑦𝑖=0 and (π‘˜1)𝑦𝑖=0, (𝑖=1,2) are satisfied. Hence, there are no more conditions for the compatibility, and the linearization criteria in the studied case are (3.3), (3.15), (3.19), (3.20) and (3.33).

Remark 3.1. In the case πœ†12=0,πœ†16=0 and πœ†15=0, one has π‘˜2=π‘˜1 and π‘˜3=π‘˜4=0. This case corresponds to (1.9).

Combining all derived results in the case πœ“1𝑦1β‰ 0, the following theorem is proven.

Theorem 3.2. Necessary and sufficient conditions for system (2.3) to be equivalent to a linear system (1.3) with constant matrix 𝐾 via fiber preserving transformations are as follows.(I)The conditions are (3.3), (3.15), (3.19), and (3.20), and the additional conditions are as(I.1)If πœ†12β‰ 0, then the additional conditions are (3.24), (3.25), and (3.27).(I.2)If πœ†12=0 and πœ†16β‰ 0, then the additional conditions are (3.29) and (3.31).(I.3)If πœ†12=0, πœ†16=0, and πœ†15β‰ 0, then the additional condition is (3.33).(I.4) If πœ†12=0, πœ†16=0, and πœ†15=0, then there are no additional conditions.

3.2. Case πœ“1𝑦1=0

If πœ“2𝑦2β‰ 0, then the change π‘₯=π‘₯,𝑦1=𝑦2,𝑦2=𝑦1 leads to the previous case. Hence, without loss of generality one can assume that πœ“2𝑦2=0. Equations (2.4) becomeπ‘Ž11=0,π‘Ž12=0,π‘Ž13=0,π‘Ž15=0,π‘Ž16=0,π‘Ž18=0,π‘Ž24=0,π‘Ž25=0,π‘Ž27πœ“=0,(3.34)2𝑦1𝑦1=βˆ’πœ“2𝑦1π‘Ž14,πœ“2π‘₯𝑦1=ξ‚€πœ‘π‘₯π‘₯πœ“2𝑦1βˆ’πœ‘π‘₯πœ“2𝑦1π‘Ž172πœ‘π‘₯ξ€Έ,πœ“1𝑦2𝑦2=βˆ’πœ“1𝑦2π‘Ž26,πœ“1π‘₯𝑦2=ξ‚€πœ‘π‘₯π‘₯πœ“1𝑦2βˆ’πœ‘π‘₯πœ“1𝑦2π‘Ž282πœ‘π‘₯ξ€Έ,πœ“1π‘₯π‘₯=ξ‚€πœ‘π‘₯π‘₯πœ“1π‘₯βˆ’πœ‘π‘₯3ξ€·π‘˜1πœ“1+π‘˜3πœ“2ξ€Έβˆ’πœ‘π‘₯πœ“1𝑦2π‘Ž29ξ‚πœ‘π‘₯,πœ“2π‘₯π‘₯=ξ‚€πœ‘π‘₯π‘₯πœ“2π‘₯βˆ’πœ‘π‘₯3ξ€·π‘˜2πœ“2+π‘˜4πœ“1ξ€Έβˆ’πœ‘π‘₯πœ“2𝑦1π‘Ž19ξ‚πœ‘π‘₯,(3.35) and Ξ”=βˆ’πœ‘π‘₯πœ“1𝑦2πœ“2𝑦1β‰ 0.

Comparing all mixed derivatives of third-order of the functions πœ“π‘—, 𝑗=1,2, one findsπ‘Žπ‘¦281=0,π‘Žπ‘¦282=2π‘Žπ‘₯26,π‘Žπ‘¦261=0,π‘Žπ‘¦172=0,π‘Žπ‘¦171=2π‘Žπ‘₯14,π‘Žπ‘¦142π‘˜=0,(3.36)1=ξ€·3πœ‘2π‘₯π‘₯βˆ’2πœ‘π‘₯π‘₯π‘₯πœ‘π‘₯+πœ‘π‘₯2πœ‡1ξ€Έξ€·4πœ‘π‘₯4ξ€Έ,π‘˜2=ξ€·3πœ‘2π‘₯π‘₯βˆ’2πœ‘π‘₯π‘₯π‘₯πœ‘π‘₯βˆ’πœ‘π‘₯2πœ‡2ξ€Έξ€·4πœ‘π‘₯4ξ€Έ,π‘˜3=ξ‚€βˆ’π‘Žπ‘¦291πœ“1𝑦2ξ‚ξ‚€πœ‘π‘₯2πœ“2𝑦1,π‘˜4=ξ‚€βˆ’π‘Žπ‘¦192πœ“2𝑦1ξ‚ξ‚€πœ‘π‘₯2πœ“1𝑦2,(3.37) where the coefficients πœ‡π‘› are defined through π‘Žπ‘–π‘— and their derivatives (presented in the appendix).

Equations (π‘˜π‘–)π‘₯=0, (π‘˜π‘–)𝑦𝑗=0(𝑖=1,2,3,4,𝑗=1,2), becomeπœ‘π‘₯π‘₯π‘₯π‘₯=ξ€·12πœ‘π‘₯π‘₯π‘₯πœ‘π‘₯π‘₯πœ‘π‘₯βˆ’12πœ‘3π‘₯π‘₯+2πœ‘π‘₯π‘₯πœ‘π‘₯2πœ‡2βˆ’πœ‘π‘₯3πœ‡2π‘₯ξ€Έξ€·2πœ‘π‘₯2ξ€Έ,ξ€·πœ‘(3.38)π‘₯π‘₯βˆ’πœ‘π‘₯πœ‡3ξ€Έπ‘Žπ‘¦291ξ€·πœ‘=0,π‘₯π‘₯βˆ’πœ‘π‘₯πœ‡4ξ€Έπ‘Žπ‘¦192=0,2πœ‘π‘₯π‘₯πœ‡5βˆ’πœ‘π‘₯πœ‡5π‘₯π‘Ž=0,(3.39)𝑦192π‘Ž14βˆ’π‘Žπ‘¦191𝑦2=0,π‘Žπ‘¦291π‘Ž26βˆ’π‘Žπ‘¦291𝑦2π‘Ž=0,𝑦291𝑦1+π‘Žπ‘¦291π‘Ž14=0,π‘Žπ‘¦192𝑦2+π‘Žπ‘¦192π‘Ž26π‘Ž=0,26π‘₯π‘₯βˆ’π‘Žπ‘₯26π‘Ž28+π‘Žπ‘¦262π‘Ž29βˆ’π‘Žπ‘¦292𝑦2+π‘Žπ‘¦292π‘Ž26π‘Ž=0,14π‘₯π‘₯βˆ’π‘Žπ‘₯14π‘Ž17+π‘Žπ‘¦141π‘Ž19βˆ’π‘Žπ‘¦191𝑦1+π‘Žπ‘¦191π‘Ž14=0.(3.40) Notice that (πœ‘π‘₯π‘₯π‘₯π‘₯)𝑦𝑗=0(𝑗=1,2) are satisfied. Equation (3.39) generates a further analysis of the compatibility depending on the values of π‘Žπ‘¦291, π‘Žπ‘¦192, and πœ‡5.

3.2.1. Case π‘Žπ‘¦291β‰ 0

From the first equation of (3.39), one obtains thatπœ‘π‘₯π‘₯=πœ‘π‘₯πœ‡3.(3.41) Substituting πœ‘π‘₯π‘₯ into the second and third equations of (3.39), one obtains2πœ‡5πœ‡3βˆ’πœ‡5π‘₯=0,π‘Žπ‘¦192ξ€·πœ‡3βˆ’πœ‡4ξ€Έ=0.(3.42) Substitution of πœ‘π‘₯π‘₯ into (3.38) gives2πœ‡3π‘₯π‘₯+πœ‡2π‘₯βˆ’6πœ‡3π‘₯πœ‡3βˆ’2πœ‡2πœ‡3+2πœ‡33=0.(3.43) Note that (πœ‘π‘₯π‘₯)𝑦1=0 and (πœ‘π‘₯π‘₯)𝑦2=0 are satisfied. Hence, there are no new conditions. In summary, the linearization criteria are (3.34), (3.36), (3.40), (3.42), and (3.43).

3.2.2. Case π‘Žπ‘¦291=0 and π‘Žπ‘¦192β‰ 0

From the second equation of (3.39), one obtains thatπœ‘π‘₯π‘₯=πœ‘π‘₯πœ‡4.(3.44) Substituting πœ‘π‘₯π‘₯ into the third equation of (3.39), one obtainsπœ‡5π‘₯βˆ’2πœ‡5πœ‡4=0.(3.45) Substitution of πœ‘π‘₯π‘₯ into (3.38) gives6πœ‡4π‘₯πœ‡4βˆ’πœ‡2π‘₯βˆ’2πœ‡4π‘₯π‘₯+2πœ‡2πœ‡4βˆ’2πœ‡43=0.(3.46) Note that (πœ‘π‘₯π‘₯)𝑦1=0 and (πœ‘π‘₯π‘₯)𝑦2=0 are satisfied. Hence, there are no other conditions. Thus, the linearization criteria in this case are (3.34), (3.36), (3.40), (3.45), and (3.46).

3.2.3. Case π‘Žπ‘¦291=0,π‘Žπ‘¦192=0, and πœ‡5β‰ 0

From the third equation of (3.39), one obtainsπœ‘π‘₯π‘₯=ξ€·πœ‘π‘₯πœ‡5π‘₯ξ€Έξ€·2πœ‡5ξ€Έ.(3.47) Substitution of πœ‘π‘₯π‘₯ into (3.38) leads to the conditionπœ‡52ξ€·4πœ‡5π‘₯πœ‡2βˆ’4πœ‡5π‘₯π‘₯π‘₯ξ€Έ+18πœ‡5π‘₯π‘₯πœ‡5π‘₯πœ‡5βˆ’15πœ‡5π‘₯3βˆ’4πœ‡2π‘₯πœ‡53=0.(3.48) Note that (πœ‘π‘₯π‘₯)𝑦1=0 and (πœ‘π‘₯π‘₯)𝑦2=0 are satisfied. Hence, there are no more conditions. In brief, the linearization criteria are conditions (3.34), (3.36), (3.40), and (3.48). Notice also thatπ‘˜2=π‘˜1+πœ‡5ξ€·4πœ‘π‘₯2ξ€Έ,π‘˜3=0,π‘˜4=0.(3.49)

Remark 3.3. In the case π‘Žπ‘¦291=0,π‘Žπ‘¦192=0,πœ‡5=0, one has π‘˜2=π‘˜1 and π‘˜3=π‘˜4=0. This case corresponds to (1.9).

Combining all obtained results in the case πœ“1𝑦1=0 and πœ“2𝑦2=0, the following theorem is proven.

Theorem 3.4. Necessary and sufficient conditions for system (2.3) to be equivalent to a linear system (1.3) with constant matrix 𝐾 by fiber preserving transformations are as follows.(II)The conditions are (3.34), (3.36), and (3.40), and the additional conditions are as(II.1)If π‘Žπ‘¦291β‰ 0, then the additional conditions are (3.42) and (3.43).(II.2)If π‘Žπ‘¦291=0 and π‘Žπ‘¦192β‰ 0, then the additional conditions are (3.45) and (3.46).(II.3)If π‘Žπ‘¦291=0,π‘Žπ‘¦192=0, and πœ‡5β‰ 0, then the additional condition is (3.48).(II.4)If π‘Žπ‘¦291=0,π‘Žπ‘¦192=0, and πœ‡5=0, then there are no additional conditions.

Remark 3.5. If one assumes that the conditions (II) of Theorem 3.4 are valid, then the conditions (I) of Theorem 3.2 vanish. Moreover, these conditions also imply that πœ†12=βˆ’4π‘Žπ‘¦291, πœ†15=βˆ’4π‘Žπ‘¦192, πœ†16=βˆ’πœ‡5, and the following is valid: (a) the conditions (II.1) become a particular case of the conditions (I.1); (b) the conditions (II.3) are a particular case of the conditions (I.2); (c) the conditions (II.2) with πœ‡5β‰ 0 and πœ‡5=0 form particular cases of the conditions (I.2) and (I.3), respectively. This allows to propose the conjecture that Theorem 3.2 is valid independently of the values of πœ“1𝑦1 and πœ“2𝑦2.
Notice that this conjecture is to be expected. For example, for a linearizable single second-order equation via a point transformation, the linearizable criteria combine to only two conditions, whereas during compatibility analysis, one has to study two separable cases [17].

4. Necessary Conditions of Linearization under Point Transformations

During the study presented in the previous section, several relations for linearizability for the general case of point transformations (1.11) and for the general case of the matrix 𝐾(𝑑) were noted. These relations are the necessary conditions for linearization, and they were obtained as follows. For example, assuming that πœ“1𝑦1β‰ 0, from (2.4), one obtains the derivatives 𝑣π‘₯, 𝑣𝑦𝑗, πœ‘π‘₯π‘₯, πœ‘π‘₯𝑦𝑗, πœ‘π‘¦π‘—π‘¦π‘˜, πœ“1π‘₯π‘₯, πœ“π‘™π‘₯𝑦𝑗, πœ“π‘™π‘¦π‘—π‘¦π‘˜, (𝑗,π‘˜,𝑙=1,2). Comparing the mixed derivatives of the functions 𝑣, πœ‘, πœ“1, and πœ“2, one can find the expressions of the quantitiesπœ”π‘›(𝑛=1,2,…,15),(4.1) where πœ”π‘› are expressed through π‘Žπ‘–π‘— and their derivatives (shown in the appendix). Excluding the functions 𝑣, πœ‘, πœ“1, and πœ“2 from these expressions, one obtains the conditions𝐽𝑖=0(𝑖=1,2,…,15),(4.2) where𝐽1=πœ”1πœ”11βˆ’2πœ”1πœ”9+2πœ”10πœ”2βˆ’πœ”3πœ”6,𝐽2=πœ”1πœ”5+2πœ”2πœ”6,𝐽3=6πœ”1πœ”8βˆ’2πœ”1πœ”12+10πœ”11πœ”2βˆ’20πœ”2πœ”9βˆ’5πœ”23,𝐽4=2πœ”10πœ”2βˆ’πœ”1πœ”9,𝐽5=10πœ”1πœ”7+πœ”12πœ”2βˆ’3πœ”2πœ”8,𝐽6=πœ”1πœ”8+πœ”11πœ”2βˆ’3πœ”2πœ”9,𝐽7=4πœ”1πœ”12βˆ’2πœ”1πœ”8+10πœ”23+5πœ”3πœ”5,𝐽8=10πœ”1πœ”13+πœ”12πœ”3βˆ’3πœ”3πœ”8,𝐽9=πœ”1πœ”15βˆ’πœ”10πœ”3,𝐽10=πœ”1πœ”14+πœ”11πœ”3βˆ’3πœ”3πœ”9,𝐽11=2πœ”12πœ”2βˆ’πœ”2πœ”8+5πœ”3πœ”4,𝐽12=πœ”13πœ”2βˆ’πœ”3πœ”7,𝐽13=2πœ”15πœ”2βˆ’πœ”3πœ”9,𝐽14=πœ”14πœ”2βˆ’πœ”3πœ”8,𝐽15=2πœ”1πœ”4βˆ’2πœ”2πœ”3βˆ’πœ”2πœ”5.(4.3) After obtaining these relations, one can directly check, by substituting (2.4) into (4.2), that they are satisfied for the general case of point transformations (1.11) and for the general case of the matrix 𝐾(𝑑). Notice also that using this substitution into the conditions obtained in [18, 23], one obtains that they are not satisfied unless the matrix 𝐾=0.

Thus, the following theorem can be stated.

Theorem 4.1. The conditions (4.2) are necessary for system (2.3) to be linearizable under point transformations.

5. Examples

In this section, examples demonstrating the procedure of using the linearization theorems are presented.

Example 5.1. Let us consider a system of two second-order quadratically semilinear ordinary differential equations 𝑦1ξ…žξ…žξ€·π‘¦=π‘Ž1,𝑦2ξ€Έπ‘¦ξ…ž12𝑦+2𝑏1,𝑦2ξ€Έπ‘¦ξ…ž1π‘¦ξ…ž2𝑦+𝑐1,𝑦2ξ€Έπ‘¦ξ…ž22,𝑦2ξ…žξ…žξ€·π‘¦=𝑑1,𝑦2ξ€Έπ‘¦ξ…ž12𝑦+2𝑒1,𝑦2ξ€Έπ‘¦ξ…ž1π‘¦ξ…ž2𝑦+𝑓1,𝑦2ξ€Έπ‘¦ξ…ž22.(5.1) In [18, 20], it is shown that system (5.1) is equivalent via point transformations to the simplest equations Μˆπ‘’1=0,Μˆπ‘’2=0 if and only if 𝑆𝑖=0(𝑖=1,2,3,4),(5.2) where 𝑆1=π‘Žπ‘¦2βˆ’π‘π‘¦1+π‘π‘’βˆ’π‘π‘‘,𝑆2=𝑏𝑦2βˆ’π‘π‘¦1+ξ€·π‘Žπ‘βˆ’π‘2ξ€Έ+𝑆(π‘π‘“βˆ’π‘π‘’),3=𝑑𝑦2βˆ’π‘’π‘¦1ξ€·βˆ’(π‘Žπ‘’βˆ’π‘π‘‘)βˆ’π‘‘π‘“βˆ’π‘’2ξ€Έ,𝑆4=𝑏𝑦1+𝑓𝑦1βˆ’π‘Žπ‘¦2βˆ’π‘’π‘¦2.(5.3) Application of fiber preserving transformation to system (5.1) also leads to the same conditions (5.2).

Example 5.2. Consider a nonlinear system 𝑦1ξ…žξ…ž=βˆ’π‘¦ξ…ž12βˆ’π‘¦ξ…ž22βˆ’π‘ž1,𝑦2ξ…žξ…ž=π‘ž2βˆ’2π‘¦ξ…ž1π‘¦ξ…ž2,(5.4) where π‘ž1, π‘ž2 are constant. Applying the linearization criteria obtained in [18, 23] to system (5.4), one obtains that system (5.4) is equivalent to the free particle equations via point transformations if and only if π‘ž2=0. Let us consider the case π‘ž2β‰ 0. Note that, for system (5.4), πœ†12=βˆ’4π‘ž2,πœ†14=0,πœ†15=βˆ’4π‘ž2,πœ†16=0.(5.5) Since π‘ž2β‰ 0, then πœ†12β‰ 0, and (3.21) becomes πœ‘π‘₯π‘₯=0. Taking the simplest solution πœ‘=π‘₯ of this equation and solving the compatible system of (3.5)–(3.14) for the functions πœ“1, and πœ“2, one gets the solution πœ“1=(1/2)𝑒(𝑦1βˆ’π‘¦2) and πœ“2=(1/2)𝑒(𝑦1+𝑦2). Substituting πœ‘, πœ“1 and πœ“2 into (3.18) and (3.26), one obtains π‘˜1=π‘ž1+π‘ž2 and π‘˜2=π‘ž1βˆ’π‘ž2. Thus, Theorem 3.2 guarantees that system (5.4) can be transformed to the system of linear equations: Μˆπ‘’1+π‘˜1𝑒1=0,Μˆπ‘’2+π‘˜2𝑒2=0,(5.6) and the linearizing transformation is 𝑑=π‘₯,𝑒1=12𝑒(𝑦1βˆ’π‘¦2),𝑒2=12𝑒(𝑦1+𝑦2).(5.7)

Example 5.3. A variety of applications in science and engineering such as the well-known oscillator system, the vibration of springs, and some types of conservative systems with two degrees of freedom are of the form 𝑦1ξ…žξ…ž=𝑔1(π‘₯)𝑦1+𝑔2(π‘₯)𝑦2,𝑦2ξ…žξ…ž=𝑔3(π‘₯)𝑦2+𝑔4(π‘₯)𝑦1.(5.8) For system (5.8), πœ†12=βˆ’4𝑔4,πœ†16𝑔=41βˆ’π‘”3ξ€Έ,πœ†15=βˆ’4𝑔2,πœ†21=βˆ’2𝑔3π‘₯,πœ†22=4𝑔3πœ†π‘₯16βˆ’πœ†16π‘₯π‘₯π‘₯.(5.9) Theorem 3.2 provides conditions sufficiently for system (5.8) to be reduced to a linear system with constant coefficients via a fiber preserving transformation. For example, for the oscillator system (𝑔2=0,𝑔4=0), the condition is 8πœ†316πœ†21+4πœ†216πœ†22+18πœ†16πœ†16π‘₯π‘₯πœ†π‘₯16βˆ’15πœ†π‘₯316=0,(5.10) whereas the criteria of [18, 23] are only satisfied when 𝑔1=𝑔3.

6. Conclusion

The necessary form of a linearizable system of two second-order ordinary differential equations 𝑦1ξ…žξ…ž=𝑓1(π‘₯,𝑦1,𝑦2,π‘¦ξ…ž1,π‘¦ξ…ž2), 𝑦2ξ…žξ…ž=𝑓2(π‘₯,𝑦1,𝑦2,π‘¦ξ…ž1,π‘¦ξ…ž2) via point transformations is obtained. Some other necessary conditions were also found. Necessary and sufficient conditions for a system of two second-order ordinary differential equations to be transformed to the general form of linear system with constant coefficients via fiber preserving transformations are obtained. A linear system with constant coefficients is chosen because of its simplicity in finding the general solution. On the way of establishing of main theorems, we also give an explicit procedure for constructing this linearizing transformation.

Appendix

Consider the following:πœ†1=2π‘Žπ‘¦281βˆ’2π‘Žπ‘¦272+π‘Ž17π‘Ž25βˆ’2π‘Ž18π‘Ž24βˆ’π‘Ž25π‘Ž28+2π‘Ž26π‘Ž27,πœ†2=2π‘Žπ‘¦181βˆ’2π‘Žπ‘¦172βˆ’2π‘Ž14π‘Ž18+π‘Ž15π‘Ž17βˆ’π‘Ž15π‘Ž28+2π‘Ž16π‘Ž27,πœ†3=4π‘Žπ‘₯24βˆ’2π‘Žπ‘¦271βˆ’2π‘Ž14π‘Ž27+2π‘Ž17π‘Ž24βˆ’2π‘Ž24π‘Ž28+π‘Ž25π‘Ž27,πœ†4=2π‘Žπ‘¦281βˆ’2π‘Žπ‘₯25+π‘Ž15π‘Ž27βˆ’2π‘Ž18π‘Ž24,πœ†5=2π‘Žπ‘¦282βˆ’4π‘Žπ‘₯26+2π‘Ž16π‘Ž27βˆ’π‘Ž18π‘Ž25,πœ†6=2π‘Žπ‘¦251βˆ’4π‘Žπ‘¦242+2π‘Ž14π‘Ž25βˆ’2π‘Ž15π‘Ž24+4π‘Ž24π‘Ž26βˆ’π‘Ž225,πœ†7=4π‘Žπ‘₯16βˆ’2π‘Žπ‘¦182+π‘Ž15π‘Ž18βˆ’2π‘Ž16π‘Ž17+2π‘Ž16π‘Ž28βˆ’2π‘Ž18π‘Ž26,πœ†8=4π‘Žπ‘¦161βˆ’2π‘Žπ‘¦152βˆ’4π‘Ž14π‘Ž16+π‘Ž215βˆ’2π‘Ž15π‘Ž26+2π‘Ž16π‘Ž25,πœ†9=2π‘Žπ‘¦252βˆ’4π‘Žπ‘¦261βˆ’π‘Ž15π‘Ž25+4π‘Ž16π‘Ž24,πœ†10=2π‘Žπ‘₯18π‘Ž15βˆ’8π‘Žπ‘¦161π‘Ž19βˆ’8π‘Žπ‘¦162π‘Ž29βˆ’4π‘Žπ‘₯17π‘Ž16βˆ’4π‘Ž18π‘₯𝑦2βˆ’4π‘Žπ‘₯18π‘Ž26+2π‘Žπ‘¦181π‘Ž18+2π‘Žπ‘¦182π‘Ž17+2π‘Žπ‘¦182π‘Ž28+8π‘Žπ‘¦191π‘Ž16+8π‘Žπ‘¦192𝑦2βˆ’8π‘Žπ‘¦192π‘Ž15+8π‘Žπ‘¦192π‘Ž26+4π‘Žπ‘₯26π‘Ž18+4π‘Žπ‘₯28π‘Ž16βˆ’16π‘Žπ‘¦292π‘Ž16βˆ’2π‘Ž14π‘Ž218βˆ’2π‘Ž15π‘Ž18π‘Ž28+2π‘Ž16π‘Ž217βˆ’2π‘Ž16π‘Ž228+2π‘Ž17π‘Ž18π‘Ž26+π‘Ž218π‘Ž25+2π‘Ž18π‘Ž26π‘Ž28,πœ†11=2π‘Žπ‘¦182π‘Ž27βˆ’4π‘Žπ‘¦192π‘Ž25βˆ’4π‘Ž26π‘₯π‘₯+4π‘Žπ‘₯26π‘Ž28βˆ’4π‘Žπ‘¦261π‘Ž19βˆ’4π‘Žπ‘¦262π‘Ž29+4π‘Žπ‘¦291π‘Ž16+4π‘Žπ‘¦292𝑦2βˆ’4π‘Žπ‘¦292π‘Ž26βˆ’π‘Ž15π‘Ž18π‘Ž27+2π‘Ž16π‘Ž17π‘Ž27βˆ’2π‘Ž16π‘Ž27π‘Ž28+2π‘Ž18π‘Ž26π‘Ž27,πœ†12=2π‘Žπ‘₯27βˆ’4π‘Žπ‘¦291βˆ’π‘Ž17π‘Ž27+4π‘Ž19π‘Ž24+2π‘Ž25π‘Ž29βˆ’π‘Ž27π‘Ž28,πœ†14=πœ†12ξ€·π‘Ž28βˆ’π‘Ž17ξ€Έβˆ’π‘Ž27πœ†16βˆ’2πœ†π‘₯12,πœ†15=2π‘Žπ‘₯18βˆ’4π‘Žπ‘¦192+2π‘Ž15π‘Ž19+4π‘Ž16π‘Ž29βˆ’π‘Ž17π‘Ž18βˆ’π‘Ž18π‘Ž28,πœ†16=4π‘Žπ‘¦191βˆ’2π‘Žπ‘₯17+2π‘Žπ‘₯28βˆ’4π‘Žπ‘¦292βˆ’4π‘Ž14π‘Ž19βˆ’2π‘Ž15π‘Ž29+π‘Ž217+2π‘Ž19π‘Ž25+4π‘Ž26π‘Ž29βˆ’π‘Ž228,πœ†17=2π‘Žπ‘₯19π‘Ž25βˆ’2π‘Žπ‘¦192π‘Ž27+2π‘Žπ‘₯25π‘Ž19+4π‘Žπ‘₯26π‘Ž29+2π‘Ž28π‘₯π‘₯βˆ’2π‘Žπ‘₯28π‘Ž28βˆ’4π‘Ž29π‘₯𝑦2+4π‘Žπ‘₯29π‘Ž26βˆ’2π‘Žπ‘¦291π‘Ž18+π‘Ž15π‘Ž19π‘Ž27+2π‘Ž16π‘Ž27π‘Ž29βˆ’π‘Ž17π‘Ž18π‘Ž27+2π‘Ž18π‘Ž19π‘Ž24+π‘Ž18π‘Ž25π‘Ž29βˆ’π‘Ž18π‘Ž27π‘Ž28βˆ’π‘Ž27πœ†15,πœ†18=2π‘Žπ‘¦191πœ†14+4π‘Žπ‘₯28πœ†14βˆ’10π‘Žπ‘¦292πœ†14+2πœ†14π‘₯π‘₯+2πœ†π‘₯14π‘Ž17βˆ’2πœ†π‘₯14π‘Ž28βˆ’2π‘Ž14π‘Ž19πœ†14βˆ’π‘Ž15π‘Ž29πœ†14+π‘Ž217πœ†14βˆ’π‘Ž17π‘Ž28πœ†14βˆ’3π‘Ž18π‘Ž27πœ†14+5π‘Ž19π‘Ž25πœ†14+10π‘Ž26π‘Ž29πœ†14βˆ’2π‘Ž228πœ†14,πœ†19=8π‘Žπ‘¦291πœ†14πœ†16+8πœ†π‘₯14π‘Ž27πœ†16+18πœ†π‘₯14πœ†14+8π‘Ž17π‘Ž27πœ†14πœ†16+9π‘Ž17πœ†214βˆ’8π‘Ž19π‘Ž24πœ†14πœ†16βˆ’4π‘Ž25π‘Ž29πœ†14πœ†16+4π‘Ž227πœ†14πœ†15βˆ’4π‘Ž27π‘Ž28πœ†14πœ†16βˆ’9π‘Ž28πœ†214,πœ†20=2π‘Žπ‘₯28βˆ’4π‘Žπ‘¦292βˆ’π‘Ž18π‘Ž27+2π‘Ž19π‘Ž25+4π‘Ž26π‘Ž29βˆ’π‘Ž228,πœ†21=2π‘Žπ‘₯26π‘Ž29+π‘Ž28π‘₯π‘₯βˆ’π‘Žπ‘₯28π‘Ž28βˆ’2π‘Ž29π‘₯𝑦2+2π‘Žπ‘₯29π‘Ž26,πœ†22=4π‘Žπ‘¦292πœ†π‘₯16βˆ’2π‘Žπ‘₯28πœ†π‘₯16βˆ’πœ†16π‘₯π‘₯π‘₯βˆ’4πœ†π‘₯16π‘Ž26π‘Ž29+πœ†π‘₯16π‘Ž228,πœ†23=2π‘Žπ‘₯28βˆ’4π‘Žπ‘¦292+4π‘Ž26π‘Ž29βˆ’π‘Ž228,πœ†24ξ‚€π‘Ž=3219π‘₯𝑦1βˆ’π‘Žπ‘₯19π‘Ž14ξ‚ξ‚€π‘Žβˆ’16𝑦171π‘Ž19+π‘Žπ‘¦181π‘Ž29+π‘Žπ‘₯29π‘Ž15βˆ’π‘Ž14π‘Ž18π‘Ž29ξ‚ξ‚€π‘Ž+56𝑦191π‘Ž17βˆ’π‘Ž14π‘Ž17π‘Ž19ξ‚ξ‚€π‘Žβˆ’24𝑦191π‘Ž28+π‘Ž14π‘Ž19π‘Ž28+21π‘Ž328ξ‚€π‘Ž+160π‘₯26π‘Ž29βˆ’π‘Ž29π‘₯𝑦2+π‘Žπ‘₯29π‘Ž26ξ‚ξ‚€π‘Ž+12017π‘Ž26π‘Ž29βˆ’π‘Žπ‘¦292π‘Ž17+64π‘Ž28π‘₯π‘₯+48π‘Žπ‘₯28π‘Ž17βˆ’112π‘Žπ‘₯28π‘Ž28ξ‚€π‘Ž+88𝑦292π‘Ž28βˆ’π‘Ž26π‘Ž28π‘Ž29ξ‚βˆ’36π‘Ž15π‘Ž17π‘Ž29+20π‘Ž15π‘Ž28π‘Ž29+15π‘Ž317βˆ’9π‘Ž217π‘Ž28βˆ’27π‘Ž17π‘Ž228,πœ†25ξ€·πœ†=1215π‘₯π‘₯π‘Ž28βˆ’πœ†15π‘₯π‘₯π‘Ž17+πœ†π‘₯15π‘Ž15π‘Ž29ξ€Έξ‚€πœ†+24π‘₯15π‘Ž14π‘Ž19βˆ’π‘Žπ‘¦191πœ†π‘₯15ξ‚βˆ’16πœ†15π‘₯π‘₯π‘₯βˆ’32π‘Žπ‘₯28πœ†π‘₯15βˆ’9πœ†π‘₯15π‘Ž217+6πœ†π‘₯15π‘Ž17π‘Ž28+19πœ†π‘₯15π‘Ž228ξ‚€π‘Ž+88𝑦292πœ†π‘₯15βˆ’πœ†π‘₯15π‘Ž26π‘Ž29,πœ†26=ξ‚€πœ†215ξ‚€16π‘Žπ‘¦191+32π‘Žπ‘₯28βˆ’80π‘Žπ‘¦292βˆ’16π‘Ž14π‘Ž19βˆ’8π‘Ž15π‘Ž29+5π‘Ž217βˆ’2π‘Ž17π‘Ž28+80π‘Ž26π‘Ž29βˆ’19π‘Ž228ξ€Έ+4πœ†15ξ€·πœ†π‘₯15π‘Ž28βˆ’4πœ†15π‘₯π‘₯βˆ’πœ†π‘₯15π‘Ž17ξ€Έ+20πœ†π‘₯215ξ€Έ/ξ€·32πœ†215ξ€Έ,πœ†27=8π‘Žπ‘¦141π‘Ž19+4π‘Žπ‘¦151π‘Ž29+4π‘Ž17π‘₯𝑦1βˆ’4π‘Žπ‘¦171π‘Ž17+4π‘Žπ‘₯18π‘Ž24βˆ’2π‘Žπ‘¦181π‘Ž27βˆ’8π‘Žπ‘¦191𝑦1+8π‘Žπ‘¦191π‘Ž14βˆ’8π‘Žπ‘¦192π‘Ž24βˆ’4π‘Žπ‘₯24π‘Ž18βˆ’2π‘Žπ‘₯27π‘Ž15+8π‘Žπ‘¦291π‘Ž15+2π‘Ž14π‘Ž18π‘Ž27+π‘Ž15π‘Ž17π‘Ž27βˆ’2π‘Ž15π‘Ž25π‘Ž29+π‘Ž15π‘Ž27π‘Ž28+8π‘Ž16π‘Ž24π‘Ž29βˆ’4π‘Ž17π‘Ž18π‘Ž24βˆ’π‘Ž18π‘Ž25π‘Ž27,πœ†28=2π‘Žπ‘¦171π‘Ž27βˆ’8π‘Žπ‘¦191π‘Ž24βˆ’4π‘Ž24π‘₯π‘₯+4π‘Žπ‘₯24π‘Ž28βˆ’4π‘Žπ‘¦241π‘Ž19βˆ’4π‘Žπ‘¦242π‘Ž29+4π‘Žπ‘¦291𝑦1+4π‘Žπ‘¦291π‘Ž14βˆ’4π‘Žπ‘¦291π‘Ž25+4π‘Žπ‘¦292π‘Ž24βˆ’π‘Ž15π‘Ž227+2π‘Ž18π‘Ž24π‘Ž27,πœ†29=2π‘Žπ‘¦181π‘Ž27βˆ’2π‘Žπ‘¦191π‘Ž25βˆ’4π‘Žπ‘¦192π‘Ž24βˆ’4π‘Žπ‘¦242π‘Ž19βˆ’2π‘Ž25π‘₯π‘₯+2π‘Žπ‘₯25π‘Ž28βˆ’4π‘Žπ‘¦261π‘Ž29+4π‘Žπ‘¦291𝑦2+2π‘Žπ‘¦291π‘Ž15βˆ’4π‘Žπ‘¦291π‘Ž26βˆ’2π‘Ž14π‘Ž18π‘Ž27+2π‘Ž14π‘Ž19π‘Ž25+π‘Ž15π‘Ž17π‘Ž27βˆ’2π‘Ž15π‘Ž19π‘Ž24βˆ’π‘Ž15π‘Ž25π‘Ž29βˆ’π‘Ž15π‘Ž27π‘Ž28+4π‘Ž16π‘Ž24π‘Ž29+π‘Ž18π‘Ž25π‘Ž27+4π‘Ž19π‘Ž24π‘Ž26βˆ’π‘Ž19π‘Ž225,πœ‡1=2π‘Žπ‘₯28βˆ’4π‘Žπ‘¦292+4π‘Ž26π‘Ž29βˆ’π‘Ž228,πœ‡2=4π‘Žπ‘¦191βˆ’2π‘Žπ‘₯17βˆ’4π‘Ž14π‘Ž19+π‘Ž217,π‘Žπ‘¦291πœ‡3=2π‘Ž29π‘₯𝑦1+π‘Žπ‘¦291π‘Ž17βˆ’π‘Žπ‘¦291π‘Ž284,π‘Žπ‘¦192πœ‡4=2π‘Ž19π‘₯𝑦2βˆ’π‘Žπ‘¦192π‘Ž17+π‘Žπ‘¦192π‘Ž284,πœ‡5ξ€·πœ‡=βˆ’1+πœ‡2ξ€Έ,πœ”1=2π‘Žπ‘¦121βˆ’4π‘Žπ‘¦112βˆ’2π‘Ž11π‘Ž15+2π‘Ž12π‘Ž14βˆ’π‘Ž12π‘Ž25+4π‘Ž13π‘Ž24,πœ”2=2π‘Žπ‘¦122βˆ’4π‘Žπ‘¦131+4π‘Ž11π‘Ž16βˆ’π‘Ž12π‘Ž15+2π‘Ž12π‘Ž26βˆ’2π‘Ž13π‘Ž25,πœ”3=π‘Žπ‘¦151βˆ’2π‘Žπ‘¦142βˆ’π‘Žπ‘¦252+2π‘Žπ‘¦261βˆ’4π‘Ž11π‘Ž18+2π‘Ž12π‘Ž17βˆ’2π‘Ž12π‘Ž28+4π‘Ž13π‘Ž27,πœ”4=2π‘Žπ‘¦152βˆ’4π‘Žπ‘₯13βˆ’4π‘Žπ‘¦161βˆ’2π‘Ž13π‘Ž17βˆ’2π‘Ž13π‘Ž28+4π‘Ž14π‘Ž16βˆ’π‘Ž215+