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

Abstract

The necessary form of a linearizable system of two second-order ordinary differential equations 𝑦^1=𝑓₁(𝑥,𝑦₁,𝑦₂,𝑦^1,𝑦^2), 𝑦^2=𝑓₂(𝑥,𝑦₁,𝑦₂,𝑦^1,𝑦^2) 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 𝑛≥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 is𝑣̈+𝐶𝑣̇+𝐷𝑣+𝐸=0,(1.1) where 𝑣=𝑣(𝑡) and 𝐸=𝐸(𝑡) are vectors, 𝐶=𝐶(𝑡) and 𝐷=𝐷(𝑡) are 𝑛×𝑛 square matrices. It can be shown [19] that there exists a change 𝑢=𝑈𝑣 such that system (1.1) is reduced to one of the following forms: either 𝑢̈+𝐾₁𝑢̇=0(1.2) or𝑢̈+𝐾𝑢=0.(1.3) Here 𝑈(𝑡), 𝐾=𝐾(𝑡), and 𝐾₁=𝐾₁(𝑡) 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:𝐾=⎛⎜⎜⎝𝑎00𝑏⎞⎟⎟⎠,𝐾=⎛⎜⎜⎝𝑎0𝑎1⎞⎟⎟⎠,𝐾=⎛⎜⎜⎝−𝑎𝑐𝑎𝑐⎞⎟⎟⎠,(1.4) where 𝑎=𝑎(𝑡), 𝑏=𝑏(𝑡), and 𝑐=𝑐(𝑡)≠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) is𝑢̈₁+𝑘₁𝑢₁+𝑘₃𝑢₂=0,𝑢̈₂+𝑘₄𝑢₁+𝑘₂𝑢₂=0,(1.5) where 𝑘_𝑖, (𝑖=1,2,3,4) are constant. If 𝑘₃≠0, then finding 𝑢₂ from the first equation of (1.5) and substituting it into the second equation of (1.5), one obtains a fourth-order ordinary differential equation𝑢¹⁽⁴⁾+𝑘₁+𝑘₂𝑢̈₁+𝑘₁𝑘₂−𝑘₃𝑘₄𝑢₁=0.(1.6) Here,𝐾=⎛⎜⎜⎝𝑘¹𝑘₃𝑘₄𝑘₂⎞⎟⎟⎠.(1.7) The general solution of the last equation depends on the roots 𝜆 of the characteristic equation𝜆⁴+𝑘₁+𝑘₂𝜆²+𝑘₁𝑘₂−𝑘₃𝑘₄=0.(1.8) The solution is similar for 𝑘₄≠0. On the other hand, if 𝑘₃=0 and 𝑘₄=0, then system (1.1) is decoupled:𝑢̈₁+𝑘₁𝑢₁=0,𝑢̈₂+𝑘₂𝑢₂=0.(1.9) Notice also that if in this case 𝑘₁=𝑘₂, then the last system of equations is equivalent to the system of two trivial equations 𝑧^=0.

1.3. The Problem Solved in the Present Paper

The present paper deals with a system of two second-order ordinary differential equations𝑦^1=𝐹₁𝑥,𝑦₁,𝑦₂,𝑦^1,𝑦^2,𝑦^2=𝐹₂𝑥,𝑦₁,𝑦₂,𝑦^1,𝑦^2.(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 transformations𝑡=𝜑𝑥,𝑦₁,𝑦₂,𝑢₁=𝜓₁𝑥,𝑦₁,𝑦₂,𝑢₂=𝜓₂𝑥,𝑦₁,𝑦₂(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 transformation𝑡=𝜑(𝑥),𝑢₁=𝜓₁𝑥,𝑦₁,𝑦₂,𝑢₂=𝜓₂𝑥,𝑦₁,𝑦₂(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(𝑡). 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 𝑢^1=𝑔₁𝑥,𝑦₁,𝑦₂,𝑦^1,𝑦^2=𝐷^𝑥𝜓₁𝐷_𝑥𝜑,𝑢^1=𝐷^𝑥𝑔₁𝑢_2𝐷_𝑥𝜑,=𝑔₂𝑥,𝑦₁,𝑦₂,𝑦^1,𝑦^2=𝐷^𝑥𝜓₂𝐷_𝑥𝜑,𝑢^2=𝐷^𝑥𝑔₂𝐷_𝑥𝜑,(2.1) where𝐷_𝑥=𝜕𝜕𝑥+𝑦^1𝜕𝜕𝑦₁+𝑦^2𝜕𝜕𝑦₂+𝑦^1𝜕𝜕𝑦^1+𝑦^2𝜕𝜕𝑦^2.(2.2) Replacing 𝑢^1, 𝑢^1, 𝑢^2, and 𝑢^2 in system (1.3), it becomes𝑦𝑦^1_2=𝑦^1𝑎¹¹𝑦^12+𝑎₁₂𝑦^1𝑦^2+𝑎₁₃𝑦^22+𝑎¹⁴𝑦^12+𝑎₁₅𝑦^1𝑦^2+𝑎₁₆𝑦^22+𝑎₁₇𝑦^1+𝑎₁₈𝑦^2+𝑎₁₉,=𝑦^2𝑎¹¹𝑦^12+𝑎₁₂𝑦^1𝑦^2+𝑎₁₃𝑦^22+𝑎²⁴𝑦^12+𝑎₂₅𝑦^1𝑦^2+𝑎₂₆𝑦^22+𝑎₂₇𝑦^1+𝑎₂₈𝑦^2+𝑎₂₉,(2.3) where the coefficients 𝑎_𝑖𝑗 are expressed through the functions 𝜑, 𝜓₁, 𝜓₂, their partial derivatives, and the entries of the matrix 𝐾=(𝑘_𝑖𝑗(𝑡)) as follows:𝑎₁₁=ℎ¹𝜓_1𝑦1𝑦1+ℎ₂𝜓_2𝑦1𝑦1−𝑣𝜑_𝑦1𝑦1+𝑓₁𝜑_𝑦1³−𝑓₂𝜑_𝑦1²𝜑_𝑦2𝑎₁₂Δ,=2ℎ¹𝜓_1𝑦1𝑦2+ℎ₂𝜓_2𝑦1𝑦2−𝑣𝜑_𝑦1𝑦2+𝑓₁𝜑_𝑦1²𝜑_𝑦2−𝑓₂𝜑_𝑦1𝜑_𝑦2²𝑎₁₃Δ,=ℎ¹𝜓_1𝑦2𝑦2+ℎ₂𝜓_2𝑦2𝑦2−𝑣𝜑_𝑦2𝑦2+𝑓₁𝜑_𝑦1𝜑_𝑦2²−𝑓₂𝜑_𝑦2³𝑎₁₄Δ,=2ℎ¹𝜓_1𝑥𝑦1+2ℎ₂𝜓_2𝑥𝑦1+ℎ₃𝜓_1𝑦1𝑦1+ℎ₄𝜓_2𝑦1𝑦1+ℎ₅𝜑_𝑦1𝑦1−2𝑣𝜑_𝑥𝑦1+3𝑓₁𝜑_𝑥𝜑_𝑦1²−2𝑓₂𝜑_𝑥𝜑_𝑦1𝜑_𝑦2−𝔍𝑎₁₅Δ,=2ℎ¹𝜓_1𝑥𝑦2+ℎ₂𝜓_2𝑥𝑦2+ℎ₃𝜓_1𝑦1𝑦2+ℎ₄𝜓_2𝑦1𝑦2+ℎ₅𝜑_𝑦1𝑦2−2𝑣𝜑_𝑥𝑦2+2𝑓₁𝜑_𝑥𝜑_𝑦1𝜑_𝑦2−𝑓₂𝜑_𝑥𝜑_𝑦2²−ℜ𝑎₁₆Δ,=ℎ³𝜓_1𝑦2𝑦2+ℎ₄𝜓_2𝑦2𝑦2+ℎ₅𝜑_𝑦2𝑦2+𝑓₁𝜑_𝑥𝜑_𝑦2²−𝑓₃𝜑_𝑦2³𝑎₁₇Δ,=ℎ¹𝜓_1𝑥𝑥+ℎ₂𝜓_2𝑥𝑥+2ℎ₃𝜓_1𝑥𝑦1+2ℎ₄𝜓_2𝑥𝑦1+2ℎ₅𝜑_𝑥𝑦1−𝑣𝜑_𝑥𝑥+3𝑓₁𝜑_𝑥2𝜑_𝑦1−𝑓₂𝜑_𝑥2𝜑_𝑦2−2𝑓₃𝜑_𝑥𝜑_𝑦1𝑎₁₈Δ,=2ℎ³𝜓_1𝑥𝑦2+ℎ₄𝜓_2𝑥𝑦2+2ℎ₅𝜑_𝑥𝑦2+𝑓₁𝜑_𝑥2𝜑_𝑦2−𝑓₃𝜑_𝑥𝜑_𝑦2²𝑎₁₉Δ,=ℎ³𝜓_1𝑥𝑥+ℎ₄𝜓_2𝑥𝑥+ℎ₅𝜑_𝑥𝑥+𝑓₁𝜑_𝑥3−𝑓₃𝜑_𝑥2𝜑_𝑦2𝑎₂₄Δ,=ℎ⁶𝜓_1𝑦1𝑦1+ℎ₇𝜓_2𝑦1𝑦1+ℎ₈𝜑_𝑦1𝑦1−𝑓₂𝜑_𝑥𝜑_𝑦1²+𝑓₃𝜑_𝑦1³𝑎₂₅Δ,=2ℎ¹𝜓_1𝑥𝑦1+ℎ₂𝜓_2𝑥𝑦1−𝑣𝜑_𝑥𝑦1+ℎ₆𝜓_1𝑦1𝑦2+ℎ₇𝜓_2𝑦1𝑦2+𝑓₁𝜑_𝑥𝜑_𝑥𝜑_𝑦1²−2𝑓₂𝜑_𝑥𝜑_𝑦1𝜑_𝑦2−𝑓₃𝜑_𝑦1²𝜑_𝑦2𝑎₂₆Δ,=2ℎ¹𝜓_1𝑥𝑦2+2ℎ₂𝜓_2𝑥𝑦2−2𝑣𝜑_𝑥𝑦2+ℎ₆𝜓_1𝑦2𝑦2+ℎ₇𝜓_2𝑦2𝑦2+ℎ₈𝜑_𝑦2𝑦2+2𝑓₁𝜑_𝑥𝜑_𝑦1𝜑_𝑦2−3𝑓₂𝜑_𝑥𝜑_𝑦2²−ℜ𝑎₂₇Δ,=2ℎ⁶𝜓_1𝑥𝑦1+ℎ₇𝜓_2𝑥𝑦1+ℎ₈𝜑_𝑥𝑦1−𝑓₂𝜑_𝑥2𝜑_𝑦1+𝑓₃𝜑_𝑥𝜑_𝑦1²𝑎₂₈Δ,=ℎ¹𝜓_1𝑥𝑥+ℎ₂𝜓_2𝑥𝑥−𝑣𝜑_𝑥𝑥+2ℎ₆𝜓_1𝑥𝑦2+2ℎ₇𝜓_2𝑥𝑦2+2ℎ₈𝜑_𝑥𝑦2+𝑓₁𝜑_𝑥2𝜑_𝑦1−3𝑓₂𝜑_𝑥2𝜑_𝑦2+𝔘𝑎₂₉Δ,=ℎ⁶𝜓_1𝑥𝑥+ℎ₇𝜓_2𝑥𝑥+ℎ₈𝜑_𝑥𝑥−𝑓₂𝜑_𝑥3+𝑓₃𝜑_𝑥2𝜑_𝑦1Δ,(2.4) where 𝔍 denotes 𝑓₃𝜑_𝑦1²𝜑_𝑦2, ℜ denotes 𝑓₃𝜑_𝑦1𝜑_𝑦2², 𝔘 denotes 2𝑓₃𝜑_𝑥𝜑_𝑦1𝜑_𝑦2, and Δ≠0 is the Jacobian of the change of variables  (1.11)Δ=𝜑_𝑥𝜓_1𝑦1𝜓_2𝑦2−𝑓𝜑_𝑥1𝜓_1𝑦2𝜓_2𝑦1−𝜑_𝑦1𝜓_1𝑥𝜓_2𝑦2+𝜑_𝑦1𝜓_1𝑦2𝜓_2𝑥+𝜑_𝑦2𝜓_1𝑥𝜓_2𝑦1−𝜑_𝑦2𝜓_1𝑦1𝜓_2𝑥,𝑓₂=𝜓_1𝑦2𝑘₂₂𝜓₂+𝑘₂₁𝜓₁−𝜓_2𝑦2𝑘₁₁𝜓₁+𝑘₁₂𝜓₂,𝑓₃=𝜓_1𝑦1𝑘₂₂𝜓₂+𝑘₂₁𝜓₁−𝜓_2𝑦1𝑘₁₁𝜓₁+𝑘₁₂𝜓₂,𝑣=𝜓_1𝑦2=𝜓_1𝑥𝑘₂₂𝜓₂+𝑘₂₁𝜓₁−𝜓_2𝑥𝑘₁₁𝜓₁+𝑘₁₂𝜓₂,ℎ₃𝜓_2𝑦1−𝜓_1𝑦1𝜓_2𝑦2,ℎ₁=𝜑_𝑦2𝜓_2𝑦1−𝜑_𝑦1𝜓_2𝑦2,ℎ₂=𝜑_𝑦1𝜓_1𝑦2−𝜑_𝑦2𝜓_1𝑦1,ℎ₆=𝜑_𝑦2𝜓_2𝑥−𝜑_𝑥𝜓_2𝑦2,ℎ₄=𝜑_𝑥𝜓_1𝑦2−𝜑_𝑦2𝜓_1𝑥,ℎ₅=𝜓_1𝑥𝜓_2𝑦2−𝜓_1𝑦2𝜓_2𝑥,=𝜑_𝑥𝜓_2𝑦1−𝜑_𝑦1𝜓_2𝑥,ℎ₇=𝜑_𝑦1𝜓_1𝑥−𝜑_𝑥𝜓_1𝑦1,ℎ₈=𝜓_1𝑦1𝜓_2𝑥−𝜓_1𝑥𝜓_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 𝜓₁, 𝜓₂, and 𝜑 with given coefficients 𝑎_𝑖𝑗 of system (2.3). The compatibility analysis depends on the value of 𝜓_1𝑦1.

The next part of the present paper deals with a fiber preserving set of point transformations (1.12):𝑡=𝜑(𝑥),𝑢₁=𝜓₁𝑥,𝑦₁,𝑦₂,𝑢₂=𝜓₂𝑥,𝑦₁,𝑦₂,(3.1) and constant matrix𝐾=⎛⎜⎜⎝𝑘¹𝑘₃𝑘₄𝑘₂⎞⎟⎟⎠.(3.2)

3.1. Case 𝜓_1𝑦1≠0

Substitution of 𝜑_𝑦1=0 and 𝜑_𝑦2=0 into (2.4) gives 𝑣𝑎₁₁=0,𝑎₁₂=0,𝑎₁₃=0,(3.3)_𝑥=2𝜑^𝑥𝑥𝑣−𝜑_𝑥𝑣𝑎₁₇+𝑎₂₈𝑣_𝑦12𝜑_𝑥,=−𝑣2𝑎¹⁴+𝑎₂₅𝑣_𝑦22,=−𝑣𝑎¹⁵+2𝑎₂₆𝜓2,(3.4)_1𝑥𝑥=𝜑^𝑥𝑥𝜓_1𝑥−𝜑_𝑥3𝑘₁𝜓₁+𝑘₃𝜓₂−𝜑_𝑥𝜓_1𝑦1𝑎₁₉−𝜑_𝑥𝜓_1𝑦2𝑎₂₉𝜑_𝑥𝜓,(3.5)_1𝑥𝑦1=𝜑^𝑥𝑥𝜓_1𝑦1−𝜑_𝑥𝜓_1𝑦1𝑎₁₇−𝜑_𝑥𝜓_1𝑦2𝑎₂₇𝜓2𝜑_𝑥,(3.6)_1𝑥𝑦2=𝜑^𝑥𝑥𝜓_1𝑦2−𝜑_𝑥𝜓_1𝑦1𝑎₁₈−𝜑_𝑥𝜓_1𝑦2𝑎₂₈𝜓2𝜑_𝑥,(3.7)_1𝑦1𝑦1𝜓=−𝜓_1𝑦1𝑎₁₄+𝜓_1𝑦2𝑎₂₄,(3.8)_1𝑦1𝑦2=−𝜓^1𝑦1𝑎₁₅+𝜓_1𝑦2𝑎₂₅𝜓2,(3.9)_1𝑦2𝑦2𝜓=−𝜓_1𝑦1𝑎₁₆+𝜓_1𝑦2𝑎₂₆,(3.10)_2𝑦2=𝜓^1𝑦2𝜓_2𝑦1−𝑣𝜓_1𝑦1𝜓,(3.11)_2𝑦1𝑦1=𝑎²⁴𝑣−𝜓_1𝑦1𝜓_2𝑦1𝑎₁₄−𝜓_1𝑦2𝜓_2𝑦1𝑎₂₄𝜓_1𝑦1𝜓,(3.12)_2𝑥𝑦1=𝜑^𝑥𝑥𝜓_1𝑦1𝜓_2𝑦1−𝜑_𝑥𝜓_1𝑦1𝜓_2𝑦1𝑎₁₇−𝜑_𝑥𝜓_1𝑦2𝜓_2𝑦1𝑎₂₇+𝜑_𝑥𝑎₂₇𝑣𝜓2𝜑^𝑥𝜓_1𝑦1,(3.13)_2𝑥𝑥=𝜑^𝑥𝑥𝜓_1𝑦1𝜓_2𝑥−𝜑_𝑥3𝜓_1𝑦1𝑘₂𝜓₂+𝑘₄𝜓₁−𝜑_𝑥𝜓_1𝑦1𝜓_2𝑦1𝑎₁₉−𝜑_𝑥𝜓_1𝑦2𝜓_2𝑦1𝑎₂₉+𝜑_𝑥𝑎₂₉𝑣𝜑^𝑥𝜓_1𝑦1,(3.14) where Δ=−𝜑_𝑥𝑣≠0. Comparing the mixed derivatives (𝑣_𝑥)_𝑦1=(𝑣_𝑦1)_𝑥,(𝑣_𝑥)_𝑦2=(𝑣_𝑦2)_𝑥 and (𝑣_𝑦2)_𝑦1=(𝑣_𝑦1)_𝑦2, one obtains 2𝑎_𝑥14−𝑎_𝑦171+𝑎𝑎_𝑥25−𝑎_𝑦281=0,𝑎_𝑥15−𝑎_𝑦172+2𝑎_𝑥26−𝑎_𝑦282=0,_𝑦151−2𝑎_𝑦142−𝑎_𝑦252+2𝑎_𝑦261=0.(3.15) Considering the conditions (𝜓_𝑖𝑥𝑥)_𝑦𝑗=(𝜓_{𝑖𝑥𝑦𝑗})_𝑥(𝑖,𝑗=1,2), one has 𝜑_𝑥𝑥𝑥=3𝜑^2𝑥𝑥𝜓_1𝑦1−4𝜑_𝑥4𝜓_1𝑦1𝑘₁−4𝜑_𝑥4𝜓_2𝑦1𝑘₃+𝜑_𝑥2𝜓_1𝑦1𝜆₂₀−𝜆₁₆+𝜑_𝑥2𝜓_1𝑦2𝜆₁₂𝑘2𝜑^𝑥𝜓_1𝑦1,(3.16)₄=4𝜑^𝑥2𝜓_1𝑦1𝜓_2𝑦1𝑘₁−𝑘₂+4𝜑_𝑥2𝜓_2𝑦1²𝑘₃−𝑣𝜆₁₂𝑘4𝜑^𝑥2𝜓_1𝑦1²,(3.17)₃=𝜓^1𝑦22𝜆₁₂−𝜓_1𝑦1²𝜆₁₅−𝜓_1𝑦1𝜓_1𝑦2𝜆₁₆𝑘₂4𝜑_𝑥2𝑣,=4𝜑^𝑥2𝜓_1𝑦1𝑘₁𝑣−2𝜓_1𝑦1²𝜓_2𝑦1𝜆₁₅−2𝜓_1𝑦1𝜓_1𝑦2𝜓_2𝑦1𝜆₁₆+𝜓_1𝑦1𝑣𝜆₁₆+2𝜓_1𝑦2²𝜓_2𝑦1𝜆₁₂−2𝜓_1𝑦2𝑣𝜆₁₂4𝜑^𝑥2𝜓_1𝑦1𝑣(3..18) Here, the functions 𝜆_𝑛(𝑥,𝑦₁,𝑦₂) are defined through 𝑎_𝑖𝑗(𝑥,𝑦₁,𝑦₂) and their derivatives (presented in the appendix).

Equating the mixed derivatives (𝜓_{𝑖𝑥𝑦1})_𝑦2=(𝜓_{𝑖𝑥𝑦2})_𝑦1, (𝜓_{𝑖𝑥𝑦1})_𝑦1=(𝜓_{𝑖𝑦1𝑦1})_𝑥, (𝜓_1𝑥𝑦2)_𝑦𝑖=(𝜓_{1𝑦𝑖𝑦2})_𝑥, (𝜓_{1𝑦1𝑦𝑖})_𝑦2=(𝜓_{1𝑦𝑖𝑦2})_𝑦1, (𝜓_2𝑦1𝑦1)_𝑦2=(𝜓_2𝑦2)_𝑦1𝑦1 and using the conditions (𝑘_𝑖+1)_𝑦2=0, (𝑖=1,2), one obtains𝜆_𝑛=0(𝑛=1,2,…,11).(3.19) Note that (𝜓_1𝑥𝑦2)_𝑦1−(𝜓_1𝑦1𝑦2)_𝑥=0 is satisfied. Differentiating (3.16) and (3.17) with respect to 𝑦₁ and 𝑦₂, one has2𝜆_𝑦151−2𝑎₁₄𝜆₁₅−𝑎₁₅𝜆₁₆+𝑎₂₅𝜆₁₅=0,𝜆_27+𝑗=0(𝑗=0,1,2).(3.20)

Equations (𝑘_𝑖)_𝑥=0(𝑖=2,3,4), become 2𝜑_𝑥𝑥4𝜑_𝑥𝑥𝜆₁₂+𝜑_𝑥𝜆₁₄=0,(3.21)4𝜑_𝑥𝑥𝜆₁₆−𝜑_𝑥𝜆_𝑥16+𝑎₁₈𝜆₁₂−𝑎₂₇𝜆₁₅=0,(3.22)𝜆₁₅+𝜑_𝑥𝑎₁₇𝜆₁₅−2𝜆_𝑥15+𝑎₁₈𝜆₁₆−𝑎₂₈𝜆₁₅=0.(3.23) Further analysis of the compatibility depends on the values of the coefficients 𝜆₁₂, 𝜆₁₆ and 𝜆₁₅ of the last three equations (3.21)–(3.23).

3.1.1. Case 𝜆₁₂≠0

Substituting 𝜑_𝑥𝑥, found from (3.21), into (3.23) and (3.22), one obtains𝑎₁₇𝜆₁₂2𝜆𝜆_15𝑥16−2𝜆_𝑥15𝜆₁₂+𝑎₁₈𝜆₁₂𝜆₁₆−𝑎₂₈𝜆₁₂𝜆₁₅−𝜆₁₄𝜆₁₅=0,𝜆₁₂+2𝑎₁₈𝜆₂₁₂−2𝑎₂₇𝜆₁₂𝜆₁₅+𝜆₁₄𝜆₁₆=0.(3.24) Differentiating (3.21) with respect to 𝑦₁ and 𝑦₂, one has𝜆_𝑦121𝜆₁₄−𝜆_𝑦141𝜆₁₂=0,𝜆_𝑦122𝜆₁₄−𝜆_𝑦142𝜆₁₂=0.(3.25) Equation (3.16) becomes𝑘₁=16𝜓_1𝑦1²𝜓_2𝑦1𝜆₂₁₂𝜆₁₅+16𝜓_1𝑦1𝜓_1𝑦2𝜓_2𝑦1𝜆₂₁₂𝜆₁₆++𝜓16𝜓_1𝑦11𝑦1𝜆₂₁₂𝑣𝜆₂₀−𝜆₁₆+4𝜓_1𝑦1𝜆₁₂𝑣2𝜆_𝑥14+𝑎₁₇𝜆₁₄−𝑎₂₈𝜆₁₄𝜆₁₄𝑣4𝑎₂₇𝜆₁₆+5𝜆₁₄−16𝜓_1𝑦2²𝜓_2𝑦1𝜆₃₁₂+16𝜓_1𝑦2𝜆₃₁₂𝑣/64𝜑_𝑥2𝜓_1𝑦1𝜆₂₁₂𝑣.(3.26) Differentiating (3.26) with respect to 𝑥, one gets the condition32𝜆₃₁₂𝜆₁₇+8𝜆₂₁₂𝜆₁₈+2𝜆₁₂𝜆₁₉+𝜆₁₄8𝑎₂₂₇𝜆₂₁₆+18𝑎₂₇𝜆₁₄𝜆₁₆+15𝜆₂₁₄=0.(3.27) Notice that (𝑘₁)_𝑦1=0 and (𝑘₁)_𝑦2=0 are satisfied. Hence, there are no new conditions for the functions 𝜑(𝑥), 𝜓₁(𝑥,𝑦₁,𝑦₂), and 𝜓₂(𝑥,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 𝜆₁₂=0 and 𝜆₁₆≠0

Since 𝜆₁₂=0 and 𝜑_𝑥≠0, (3.21) leads to the condition 𝜆₁₄=0, and (3.22) becomes𝜑_𝑥𝑥=𝜑^𝑥𝜆_𝑥162𝜆₁₆.(3.28) Substituting 𝜑_𝑥𝑥 into (3.23) and (3.16), one gets𝑘2𝜆_𝑥16𝜆₁₅−𝜆_𝑥15𝜆₁₆+𝜆₁₅𝜆₁₆𝑎₁₇−𝑎₂₈+𝑎₁₈𝜆₂₁₆=0,(3.29)₁=4𝜓^1𝑦1𝜓_2𝑦1𝜆₂₁₆𝜆₁₅+4𝜓_1𝑦2𝜓_2𝑦1𝜆₃₁₆−4𝜆₃₁₆𝑣+4𝜆₂₁₆𝜆₂₃𝑣−4𝜆₁₆𝜆_16𝑥𝑥𝑣+5𝜆_𝑥216𝑣16𝜑_𝑥2𝜆₂₁₆𝑣.(3.30) Equation (𝑘₁)_𝑥=0 leads to the condition 8𝜆₃₁₆𝜆₂₁+4𝜆₂₁₆𝜆₂₂+18𝜆₁₆𝜆_16𝑥𝑥𝜆_𝑥16−15𝜆_𝑥316=0.(3.31) Note that (𝜑_𝑥𝑥)_𝑦𝑖=0 and (𝑘₁)_𝑦𝑖=0, (𝑖=1,2) are satisfied. Hence, there are no other conditions for the functions 𝜑(𝑥), 𝜓₁(𝑥,𝑦₁,𝑦₂), and 𝜓₂(𝑥,𝑦₁,𝑦₂). Summarizing, the linearization criteria in the case 𝜆₁₂=0 and 𝜆₁₆≠0 are conditions (3.3), (3.15), (3.19), (3.20), (3.29), and (3.31).

3.1.3. Case 𝜆₁₂=0,𝜆₁₆=0, and 𝜆₁₅≠0

Substituting 𝜑_𝑥𝑥, found from (3.23), into (3.16), one has𝑘₁=16𝜓^1𝑦1𝜓_2𝑦1𝜆₃₁₅+𝜆₂₆𝑣64𝜑_𝑥2𝜆₂₁₅𝑣.(3.32) Differentiating (3.32) with respect to 𝑥, one gets𝜆₃₁₅𝜆₂₄+2𝜆₂₁₅𝜆₂₅−120𝜆_𝑥315+36𝜆₁₅𝜆_𝑥154𝜆_15𝑥𝑥+𝜆_𝑥15𝑎₁₇−𝜆_𝑥15𝑎₂₈=0.(3.33) Note that the equations (𝜑_𝑥𝑥)_𝑦𝑖=0 and (𝑘₁)_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 𝜆₁₂=0,𝜆₁₆=0 and 𝜆₁₅=0, one has 𝑘₂=𝑘₁ and 𝑘₃=𝑘₄=0. This case corresponds to (1.9).

Combining all derived results in the case 𝜓_1𝑦1≠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 𝜆₁₂≠0, then the additional conditions are (3.24), (3.25), and (3.27).(I.2)If 𝜆₁₂=0 and 𝜆₁₆≠0, then the additional conditions are (3.29) and (3.31).(I.3)If 𝜆₁₂=0, 𝜆₁₆=0, and 𝜆₁₅≠0, then the additional condition is (3.33).(I.4) If 𝜆₁₂=0, 𝜆₁₆=0, and 𝜆₁₅=0, then there are no additional conditions.

3.2. Case 𝜓_1𝑦1=0

If 𝜓_2𝑦2≠0, then the change 𝑥=𝑥,𝑦₁=𝑦₂,𝑦₂=𝑦₁ leads to the previous case. Hence, without loss of generality one can assume that 𝜓_2𝑦2=0. Equations (2.4) become𝑎₁₁=0,𝑎₁₂=0,𝜓𝑎₁₃=0,𝑎₁₅=0,𝑎₁₆=0,𝑎₁₈=0,𝑎₂₄=0,𝑎₂₅=0,𝑎₂₇=0,(3.34)_2𝑦1𝑦1=−𝜓_2𝑦1𝑎₁₄,𝜓_2𝑥𝑦1=𝜑^𝑥𝑥𝜓_2𝑦1−𝜑_𝑥𝜓_2𝑦1𝑎₁₇𝜓_1𝑦2𝑦22𝜑_𝑥,=−𝜓_1𝑦2𝑎₂₆,𝜓_1𝑥𝑦2=𝜑^𝑥𝑥𝜓_1𝑦2−𝜑_𝑥𝜓_1𝑦2𝑎₂₈𝜓_1𝑥𝑥2𝜑_𝑥,=𝜑^𝑥𝑥𝜓_1𝑥−𝜑_𝑥3𝑘₁𝜓₁+𝑘₃𝜓₂−𝜑_𝑥𝜓_1𝑦2𝑎₂₉𝜑_𝑥𝜓_2𝑥𝑥,=𝜑^𝑥𝑥𝜓_2𝑥−𝜑_𝑥3𝑘₂𝜓₂+𝑘₄𝜓₁−𝜑_𝑥𝜓_2𝑦1𝑎₁₉𝜑_𝑥,(3.35) and Δ=−𝜑_𝑥𝜓_1𝑦2𝜓_2𝑦1≠0.

Comparing all mixed derivatives of third-order of the functions 𝜓_𝑗, 𝑗=1,2, one finds𝑎𝑘_𝑦281=0,𝑎_𝑦282=2𝑎_𝑥26,𝑎_𝑦261=0,𝑎_𝑦172=0,𝑎_𝑦171=2𝑎_𝑥14,𝑎_𝑦142=0,(3.36)₁=3𝜑^2𝑥𝑥−2𝜑_𝑥𝑥𝑥𝜑_𝑥+𝜑_𝑥2𝜇₁4𝜑_𝑥4,𝑘²=3𝜑^2𝑥𝑥−2𝜑_𝑥𝑥𝑥𝜑_𝑥−𝜑_𝑥2𝜇₂𝑘₃4𝜑_𝑥4,=−𝑎^𝑦291𝜓_1𝑦2𝜑^𝑥2𝜓_2𝑦1,𝑘⁴=−𝑎^𝑦192𝜓_2𝑦1𝜑^𝑥2𝜓_1𝑦2,(3.37) where the coefficients 𝜇_𝑛 are defined through 𝑎_𝑖𝑗 and their derivatives (presented in the appendix).

Equations (𝑘_𝑖)_𝑥=0, (𝑘_𝑖)_𝑦𝑗=0(𝑖=1,2,3,4,𝑗=1,2), become𝜑_{𝑥𝑥𝑥𝑥}=12𝜑^𝑥𝑥𝑥𝜑_𝑥𝑥𝜑_𝑥−12𝜑_3𝑥𝑥+2𝜑_𝑥𝑥𝜑_𝑥2𝜇₂−𝜑_𝑥3𝜇_2𝑥𝜑2𝜑_𝑥2,(3.38)_𝑥𝑥−𝜑_𝑥𝜇₃𝑎𝑎_𝑦291=0,𝜑_𝑥𝑥−𝜑_𝑥𝜇₄𝑎_𝑦192=0,2𝜑_𝑥𝑥𝜇₅−𝜑_𝑥𝜇_5𝑥=0,(3.39)_𝑦192𝑎𝑎₁₄−𝑎_𝑦191𝑦2=0,𝑎_𝑦291𝑎₂₆−𝑎_𝑦291𝑦2=0,_𝑦291𝑦1𝑎+𝑎_𝑦291𝑎₁₄=0,𝑎_𝑦192𝑦2+𝑎_𝑦192𝑎₂₆=0,_26𝑥𝑥𝑎−𝑎_𝑥26𝑎₂₈+𝑎_𝑦262𝑎₂₉−𝑎_𝑦292𝑦2+𝑎_𝑦292𝑎₂₆=0,_14𝑥𝑥−𝑎_𝑥14𝑎₁₇+𝑎_𝑦141𝑎₁₉−𝑎_𝑦191𝑦1+𝑎_𝑦191𝑎₁₄=0.(3.40) Notice that (𝜑_{𝑥𝑥𝑥𝑥})_𝑦𝑗=0(𝑗=1,2) are satisfied. Equation (3.39) generates a further analysis of the compatibility depending on the values of 𝑎_𝑦291, 𝑎_𝑦192, and 𝜇₅.

3.2.1. Case 𝑎_𝑦291≠0

From the first equation of (3.39), one obtains that𝜑_𝑥𝑥=𝜑_𝑥𝜇₃.(3.41) Substituting 𝜑_𝑥𝑥 into the second and third equations of (3.39), one obtains2𝜇₅𝜇₃−𝜇_5𝑥=0,𝑎_𝑦192𝜇₃−𝜇₄=0.(3.42) Substitution of 𝜑_𝑥𝑥 into (3.38) gives2𝜇_3𝑥𝑥+𝜇_2𝑥−6𝜇_3𝑥𝜇₃−2𝜇₂𝜇₃+2𝜇₃₃=0.(3.43) Note that (𝜑_𝑥𝑥)_𝑦1=0 and (𝜑_𝑥𝑥)_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 𝑎_𝑦291=0 and 𝑎_𝑦192≠0

From the second equation of (3.39), one obtains that𝜑_𝑥𝑥=𝜑_𝑥𝜇₄.(3.44) Substituting 𝜑_𝑥𝑥 into the third equation of (3.39), one obtains𝜇_5𝑥−2𝜇₅𝜇₄=0.(3.45) Substitution of 𝜑_𝑥𝑥 into (3.38) gives6𝜇_4𝑥𝜇₄−𝜇_2𝑥−2𝜇_4𝑥𝑥+2𝜇₂𝜇₄−2𝜇₄₃=0.(3.46) Note that (𝜑_𝑥𝑥)_𝑦1=0 and (𝜑_𝑥𝑥)_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 𝑎_𝑦291=0,𝑎_𝑦192=0, and 𝜇₅≠0

From the third equation of (3.39), one obtains𝜑_𝑥𝑥=𝜑^𝑥𝜇_5𝑥2𝜇₅.(3.47) Substitution of 𝜑_𝑥𝑥 into (3.38) leads to the condition𝜇₅₂4𝜇_5𝑥𝜇₂−4𝜇_{5𝑥𝑥𝑥}+18𝜇_5𝑥𝑥𝜇_5𝑥𝜇₅−15𝜇_5𝑥3−4𝜇_2𝑥𝜇₅₃=0.(3.48) Note that (𝜑_𝑥𝑥)_𝑦1=0 and (𝜑_𝑥𝑥)_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 that𝑘₂=𝑘₁+𝜇⁵4𝜑_𝑥2,𝑘³=0,𝑘₄=0.(3.49)

Remark 3.3. In the case 𝑎_𝑦291=0,𝑎_𝑦192=0,𝜇₅=0, one has 𝑘₂=𝑘₁ and 𝑘₃=𝑘₄=0. This case corresponds to (1.9).

Combining all obtained results in the case 𝜓_1𝑦1=0 and 𝜓_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 𝑎_𝑦291≠0, then the additional conditions are (3.42) and (3.43).(II.2)If 𝑎_𝑦291=0 and 𝑎_𝑦192≠0, then the additional conditions are (3.45) and (3.46).(II.3)If 𝑎_𝑦291=0,𝑎_𝑦192=0, and 𝜇₅≠0, then the additional condition is (3.48).(II.4)If 𝑎_𝑦291=0,𝑎_𝑦192=0, and 𝜇₅=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 𝜆₁₂=−4𝑎_𝑦291, 𝜆₁₅=−4𝑎_𝑦192, 𝜆₁₆=−𝜇₅, 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 𝜇₅≠0 and 𝜇₅=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 𝜓_1𝑦1 and 𝜓_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(𝑡) were noted. These relations are the necessary conditions for linearization, and they were obtained as follows. For example, assuming that 𝜓_1𝑦1≠0, from (2.4), one obtains the derivatives 𝑣_𝑥, 𝑣_𝑦𝑗, 𝜑_𝑥𝑥, 𝜑_𝑥𝑦𝑗, 𝜑_{𝑦𝑗𝑦𝑘}, 𝜓_1𝑥𝑥, 𝜓_{𝑙𝑥𝑦𝑗}, 𝜓_{𝑙𝑦𝑗𝑦𝑘}, (𝑗,𝑘,𝑙=1,2). Comparing the mixed derivatives of the functions 𝑣, 𝜑, 𝜓₁, and 𝜓₂, one can find the expressions of the quantities𝜔_𝑛(𝑛=1,2,…,15),(4.1) where 𝜔_𝑛 are expressed through 𝑎_𝑖𝑗 and their derivatives (shown in the appendix). Excluding the functions 𝑣, 𝜑, 𝜓₁, and 𝜓₂ from these expressions, one obtains the conditions𝐽_𝑖=0(𝑖=1,2,…,15),(4.2) where𝐽₁=𝜔₁𝐽𝜔₁₁₃−2𝜔₁𝜔₉+2𝜔₁₀𝜔₂−𝜔₃𝜔₆,𝐽₂=𝜔₁𝜔₅+2𝜔₂𝜔₆,𝐽₄=6𝜔₁𝜔₈−2𝜔₁𝜔₁₂+10𝜔₁₁𝜔₂−20𝜔₂𝜔₉−5𝜔²³,𝐽₆=2𝜔₁₀𝜔₂−𝜔₁𝜔₉,𝐽₅=10𝜔₁𝜔₇+𝜔₁₂𝜔₂−3𝜔₂𝜔₈,=𝜔₁𝐽𝜔₈₈+𝜔₁₁𝜔₂−3𝜔₂𝜔₉,𝐽₇=4𝜔₁𝜔₁₂−2𝜔₁𝜔₈+10𝜔²³+5𝜔₃𝜔₅,𝐽₁₀=10𝜔₁𝜔₁₃+𝜔₁₂𝜔₃−3𝜔₃𝜔₈,𝐽₉=𝜔₁𝜔₁₅−𝜔₁₀𝜔₃,=𝜔₁𝜔𝐽₁₄₁₂+𝜔₁₁𝜔₃−3𝜔₃𝜔₉,𝐽₁₁=2𝜔₁₂𝜔₂−𝜔₂𝜔₈+5𝜔₃𝜔₄,𝐽₁₄=𝜔₁₃𝜔₂−𝜔₃𝜔₇,𝐽₁₃=2𝜔₁₅𝜔₂−𝜔₃𝜔₉,=𝜔₁₄𝜔₂−𝜔₃𝜔₈,𝐽₁₅=2𝜔₁𝜔₄−2𝜔₂𝜔₃−𝜔₂𝜔₅.(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(𝑡). 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 𝑦𝑦^1_2=𝑎𝑦₁,𝑦₂𝑦^12+2𝑏𝑦₁,𝑦₂𝑦^1𝑦^2+𝑐𝑦₁,𝑦₂𝑦^22,=𝑑𝑦₁,𝑦₂𝑦^12+2𝑒𝑦₁,𝑦₂𝑦^1𝑦^2+𝑓𝑦₁,𝑦₂𝑦^22.(5.1) In [18, 20], it is shown that system (5.1) is equivalent via point transformations to the simplest equations 𝑢̈₁=0,𝑢̈₂=0 if and only if 𝑆_𝑖=0(𝑖=1,2,3,4),(5.2) where 𝑆𝑆₁=𝑎_𝑦2−𝑏_𝑦1+𝑏𝑒−𝑐𝑑,𝑆₂=𝑏_𝑦2−𝑐_𝑦1+𝑎𝑐−𝑏²+(𝑏𝑓−𝑐𝑒),₃=𝑑_𝑦2−𝑒_𝑦1−(𝑎𝑒−𝑏𝑑)−𝑑𝑓−𝑒²,𝑆₄=𝑏_𝑦1+𝑓_𝑦1−𝑎_𝑦2−𝑒_𝑦2.(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 𝑦^1=−𝑦^12−𝑦^22−𝑞₁,𝑦^2=𝑞₂−2𝑦^1𝑦^2,(5.4) where 𝑞₁, 𝑞₂ 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 𝑞₂=0. Let us consider the case 𝑞₂≠0. Note that, for system (5.4), 𝜆₁₂=−4𝑞₂,𝜆₁₄=0,𝜆₁₅=−4𝑞₂,𝜆₁₆=0.(5.5) Since 𝑞₂≠0, then 𝜆₁₂≠0, and (3.21) becomes 𝜑_𝑥𝑥=0. Taking the simplest solution 𝜑=𝑥 of this equation and solving the compatible system of (3.5)–(3.14) for the functions 𝜓₁, and 𝜓₂, one gets the solution 𝜓₁=(1/2)𝑒^{(𝑦1−𝑦2)} and 𝜓₂=(1/2)𝑒^{(𝑦1+𝑦2)}. Substituting 𝜑, 𝜓₁ and 𝜓₂ into (3.18) and (3.26), one obtains 𝑘₁=𝑞₁+𝑞₂ and 𝑘₂=𝑞₁−𝑞₂. Thus, Theorem 3.2 guarantees that system (5.4) can be transformed to the system of linear equations: 𝑢̈₁+𝑘₁𝑢₁=0,𝑢̈₂+𝑘₂𝑢₂=0,(5.6) and the linearizing transformation is 𝑡=𝑥,𝑢₁=12𝑒^{(𝑦1−𝑦2)},𝑢₂=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 𝑦^1=𝑔₁(𝑥)𝑦₁+𝑔₂(𝑥)𝑦₂,𝑦^2=𝑔₃(𝑥)𝑦₂+𝑔₄(𝑥)𝑦₁.(5.8) For system (5.8), 𝜆₁₂=−4𝑔₄,𝜆₁₆=4𝑔₁−𝑔₃,𝜆₁₅=−4𝑔₂,𝜆₂₁=−2𝑔_3𝑥,𝜆₂₂=4𝑔₃𝜆_𝑥16−𝜆_{16𝑥𝑥𝑥}.(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 8𝜆₃₁₆𝜆₂₁+4𝜆₂₁₆𝜆₂₂+18𝜆₁₆𝜆_16𝑥𝑥𝜆_𝑥16−15𝜆_𝑥316=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 𝑦^1=𝑓₁(𝑥,𝑦₁,𝑦₂,𝑦^1,𝑦^2), 𝑦^2=𝑓₂(𝑥,𝑦₁,𝑦₂,𝑦^1,𝑦^2) 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:𝜆𝜆₁₂=2𝑎_𝑦281−2𝑎_𝑦272+𝑎₁₇𝑎₂₅−2𝑎₁₈𝑎₂₄−𝑎₂₅𝑎₂₈+2𝑎₂₆𝑎₂₇,𝜆₃=2𝑎_𝑦181−2𝑎_𝑦172−2𝑎₁₄𝑎₁₈+𝑎₁₅𝑎₁₇−𝑎₁₅𝑎₂₈+2𝑎₁₆𝑎₂₇,𝜆₄=4𝑎_𝑥24−2𝑎_𝑦271−2𝑎₁₄𝑎₂₇+2𝑎₁₇𝑎₂₄−2𝑎₂₄𝑎₂₈+𝑎₂₅𝑎₂₇,𝜆₅=2𝑎_𝑦281−2𝑎_𝑥25+𝑎₁₅𝑎₂₇−2𝑎₁₈𝑎₂₄,𝜆₆=2𝑎_𝑦282−4𝑎_𝑥26+2𝑎₁₆𝑎₂₇−𝑎₁₈𝑎₂₅,𝜆₇=2𝑎_𝑦251−4𝑎_𝑦242+2𝑎₁₄𝑎₂₅−2𝑎₁₅𝑎₂₄+4𝑎₂₄𝑎₂₆−𝑎₂₂₅,𝜆₈=4𝑎_𝑥16−2𝑎_𝑦182+𝑎₁₅𝑎₁₈−2𝑎₁₆𝑎₁₇+2𝑎₁₆𝑎₂₈−2𝑎₁₈𝑎₂₆,𝜆₉=4𝑎_𝑦161−2𝑎_𝑦152−4𝑎₁₄𝑎₁₆+𝑎₂₁₅−2𝑎₁₅𝑎₂₆+2𝑎₁₆𝑎₂₅,𝜆₁₀=2𝑎_𝑦252−4𝑎_𝑦261−𝑎₁₅𝑎₂₅+4𝑎₁₆𝑎₂₄,=2𝑎_𝑥18𝑎₁₅−8𝑎_𝑦161𝑎₁₉−8𝑎_𝑦162𝑎₂₉−4𝑎_𝑥17𝑎₁₆−4𝑎_18𝑥𝑦2−4𝑎_𝑥18𝑎₂₆+2𝑎_𝑦181𝑎₁₈+2𝑎_𝑦182𝑎₁₇+2𝑎_𝑦182𝑎₂₈+8𝑎_𝑦191𝑎₁₆+8𝑎_𝑦192𝑦2−8𝑎_𝑦192𝑎₁₅+8𝑎_𝑦192𝑎₂₆+4𝑎_𝑥26𝑎₁₈+4𝑎_𝑥28𝑎₁₆−16𝑎_𝑦292𝑎₁₆−2𝑎₁₄𝑎₂₁₈−2𝑎₁₅𝑎₁₈𝑎₂₈+2𝑎₁₆𝑎₂₁₇−2𝑎₁₆𝑎₂₂₈𝜆₁₁+2𝑎₁₇𝑎₁₈𝑎₂₆+𝑎₂₁₈𝑎₂₅+2𝑎₁₈𝑎₂₆𝑎₂₈,=2𝑎_𝑦182𝑎₂₇−4𝑎_𝑦192𝑎₂₅−4𝑎_26𝑥𝑥+4𝑎_𝑥26𝑎₂₈−4𝑎_𝑦261𝑎₁₉−4𝑎_𝑦262𝑎₂₉+4𝑎_𝑦291𝑎₁₆+4𝑎_𝑦292𝑦2−4𝑎_𝑦292𝑎₂₆−𝑎₁₅𝑎₁₈𝑎₂₇+2𝑎₁₆𝑎₁₇𝑎₂₇𝜆₁₂−2𝑎₁₆𝑎₂₇𝑎₂₈+2𝑎₁₈𝑎₂₆𝑎₂₇,𝜆₁₄=2𝑎_𝑥27−4𝑎_𝑦291−𝑎₁₇𝑎₂₇+4𝑎₁₉𝑎₂₄+2𝑎₂₅𝑎₂₉−𝑎₂₇𝑎₂₈,𝜆₁₅=𝜆₁₂𝑎₂₈−𝑎₁₇−𝑎₂₇𝜆₁₆−2𝜆_𝑥12,𝜆₁₆=2𝑎_𝑥18−4𝑎_𝑦192+2𝑎₁₅𝑎₁₉+4𝑎₁₆𝑎₂₉−𝑎₁₇𝑎₁₈−𝑎₁₈𝑎₂₈,=4𝑎_𝑦191−2𝑎_𝑥17+2𝑎_𝑥28−4𝑎_𝑦292−4𝑎₁₄𝑎₁₉−2𝑎₁₅𝑎₂₉+𝑎₂₁₇𝜆₁₇+2𝑎₁₉𝑎₂₅+4𝑎₂₆𝑎₂₉−𝑎₂₂₈,=2𝑎_𝑥19𝑎₂₅−2𝑎_𝑦192𝑎₂₇+2𝑎_𝑥25𝑎₁₉+4𝑎_𝑥26𝑎₂₉+2𝑎_28𝑥𝑥−2𝑎_𝑥28𝑎₂₈−4𝑎_29𝑥𝑦2+4𝑎_𝑥29𝑎₂₆−2𝑎_𝑦291𝑎₁₈+𝑎₁₅𝑎₁₉𝑎₂₇+2𝑎₁₆𝑎₂₇𝑎₂₉−𝑎₁₇𝑎₁₈𝑎₂₇𝜆₁₈+2𝑎₁₈𝑎₁₉𝑎₂₄+𝑎₁₈𝑎₂₅𝑎₂₉−𝑎₁₈𝑎₂₇𝑎₂₈−𝑎₂₇𝜆₁₅,=2𝑎_𝑦191𝜆₁₄+4𝑎_𝑥28𝜆₁₄−10𝑎_𝑦292𝜆₁4+2𝜆_14𝑥𝑥+2𝜆_𝑥14𝑎₁₇−2𝜆_𝑥14𝑎₂₈−2𝑎₁₄𝑎₁₉𝜆₁₄−𝑎₁₅𝑎₂₉𝜆₁₄+𝑎₂₁₇𝜆₁₄−𝑎₁₇𝑎₂₈𝜆₁₄−3𝑎₁₈𝑎₂₇𝜆₁₄𝜆₁₉+5𝑎₁₉𝑎₂₅𝜆₁₄+10𝑎₂₆𝑎₂₉𝜆₁₄−2𝑎₂₂₈𝜆₁₄,=8𝑎_𝑦291𝜆₁₄𝜆₁₆+8𝜆_𝑥14𝑎₂₇𝜆₁₆+18𝜆_𝑥14𝜆₁₄+8𝑎₁₇𝑎₂₇𝜆₁₄𝜆₁₆+9𝑎₁₇𝜆₂₁₄−8𝑎₁₉𝑎₂₄𝜆₁₄𝜆₁₆−4𝑎₂₅𝑎₂₉𝜆₁₄𝜆₁₆+4𝑎₂₂₇𝜆₁₄𝜆₁₅−4𝑎₂₇𝑎₂₈𝜆₁₄𝜆₁₆𝜆₂₀−9𝑎₂₈𝜆₂₁₄,𝜆₂₁=2𝑎_𝑥28−4𝑎_𝑦292−𝑎₁₈𝑎₂₇+2𝑎₁₉𝑎₂₅+4𝑎₂₆𝑎₂₉−𝑎₂₂₈,𝜆₂₂=2𝑎_𝑥26𝑎₂₉+𝑎_28𝑥𝑥−𝑎_𝑥28𝑎₂₈−2𝑎_29𝑥𝑦2+2𝑎_𝑥29𝑎₂₆,𝜆₂₃=4𝑎_𝑦292𝜆_𝑥16−2𝑎_𝑥28𝜆_𝑥16−𝜆_{16𝑥𝑥𝑥}−4𝜆_𝑥16𝑎₂₆𝑎₂₉+𝜆_𝑥16𝑎₂₂₈,𝜆₂₄=2𝑎_𝑥28−4𝑎_𝑦292+4𝑎₂₆𝑎₂₉−𝑎₂₂₈,=+3256𝑎𝑎^19𝑥𝑦_1𝑦191−𝑎_𝑥19𝑎₁₄−16𝑎^𝑦17₁𝑎₁₉+𝑎_𝑦181𝑎₂₉+𝑎_𝑥29𝑎₁₅−𝑎₁₄𝑎₁₈𝑎₂₉𝑎₁₇−𝑎₁₄𝑎₁₇𝑎₁₉−24𝑎^𝑦19₁𝑎₂₈+𝑎₁₄𝑎₁₉𝑎₂₈+21𝑎³²⁸++61640𝑎𝑎_{𝑥2628𝑥𝑥}𝑎₂₉−𝑎_29𝑥𝑦2+𝑎_𝑥29𝑎₂₆+120𝑎¹⁷𝑎₂₆𝑎₂₉−𝑎_𝑦292𝑎₁₇−36𝑎₁₅+48𝑎_𝑥28𝑎₁₇−112𝑎_𝑥28𝑎₂₈+88𝑎_𝑦292𝑎₂₈−𝑎₂₆𝑎₂₈𝑎₂₉𝜆₂₅𝑎₁₇𝑎₂₉+20𝑎₁₅𝑎₂₈𝑎₂₉+15𝑎₃₁₇−9𝑎₂₁₇𝑎₂₈−27𝑎₁₇𝑎₂₂₈,=−1216𝜆𝜆^15𝑥𝑥𝑎_{2815𝑥𝑥𝑥}−𝜆_15𝑥𝑥𝑎₁₇+𝜆_𝑥15𝑎₁₅𝑎₂₉+24𝜆_𝑥15𝑎₁₄𝑎₁₉−𝑎_𝑦191𝜆_𝑥15−32𝑎_𝑥28𝜆_𝑥15−9𝜆_𝑥15𝑎₂₁₇+6𝜆_𝑥15𝑎₁₇𝑎₂₈+19𝜆_𝑥15𝑎₂₂₈𝜆₂₆+88𝑎_𝑦292𝜆_𝑥15−𝜆_𝑥15𝑎₂₆𝑎₂₉,=𝜆₂₁₅16𝑎^𝑦19₁+32𝑎_𝑥28−80𝑎_𝑦292−16𝑎₁₄𝑎₁₉+4𝜆−₁₅8𝑎₁₅𝑎₂₉+5𝑎₂₁₇−2𝑎₁₇𝑎₂₈+80𝑎₂₆𝑎₂₉−19𝑎₂₂₈𝜆₂₇𝜆_𝑥15𝑎₂₈−4𝜆_15𝑥𝑥−𝜆_𝑥15𝑎₁₇+20𝜆_𝑥215/32𝜆₂₁₅,=8𝑎_𝑦141𝑎₁₉+4𝑎_𝑦151𝑎₂₉+4𝑎_17𝑥𝑦1−4𝑎_𝑦171𝑎₁₇+4𝑎_𝑥18𝑎₂₄−2𝑎_𝑦181𝑎₂₇−8𝑎_𝑦191𝑦1+8𝑎_𝑦191𝑎₁₄−8𝑎_𝑦192𝑎₂₄−4𝑎_𝑥24𝑎₁₈−2𝑎_𝑥27𝑎₁₅+8𝑎_𝑦291𝑎₁₅+2𝑎₁₄𝑎₁₈𝑎₂₇+𝑎₁₅𝑎₁₇𝑎₂₇−2𝑎₁₅𝑎₂₅𝑎₂₉+𝑎₁₅𝑎₂₇𝑎₂₈+8𝑎₁₆𝑎₂₄𝑎₂₉𝜆₂₈−4𝑎₁₇𝑎₁₈𝑎₂₄−𝑎₁₈𝑎₂₅𝑎₂₇,=2𝑎_𝑦171𝑎₂₇−8𝑎_𝑦191𝑎₂₄−4𝑎_24𝑥𝑥+4𝑎_𝑥24𝑎₂₈−4𝑎_𝑦241𝑎₁₉−4𝑎_𝑦242𝑎₂₉+4𝑎_𝑦291𝑦1+4𝑎_𝑦291𝑎₁₄−4𝑎_𝑦291𝑎₂₅+4𝑎_𝑦292𝑎₂₄𝜆₂₉−𝑎₁₅𝑎₂₂₇+2𝑎₁₈𝑎₂₄𝑎₂₇,=2𝑎_𝑦181𝑎₂₇−2𝑎_𝑦191𝑎₂₅−4𝑎_𝑦192𝑎₂₄−4𝑎_𝑦242𝑎₁₉−2𝑎_25𝑥𝑥+2𝑎_𝑥25𝑎₂₈−4𝑎_𝑦261𝑎₂₉+4𝑎_𝑦291𝑦2+2𝑎_𝑦291𝑎₁₅−4𝑎_𝑦291𝑎₂₆−2𝑎₁₄𝑎₁₈𝑎₂₇+2𝑎₁₄𝑎₁₉𝑎₂₅+𝑎₁₅𝑎₁₇𝑎₂₇−2𝑎₁₅𝑎₁₉𝑎₂₄−𝑎₁₅𝑎₂₅𝑎₂₉−𝑎₁₅𝑎₂₇𝑎₂₈𝜇₁+4𝑎₁₆𝑎₂₄𝑎₂₉+𝑎₁₈𝑎₂₅𝑎₂₇+4𝑎₁₉𝑎₂₄𝑎₂₆−𝑎₁₉𝑎₂₂₅,𝜇₂=2𝑎_𝑥28−4𝑎_𝑦292+4𝑎₂₆𝑎₂₉−𝑎₂₂₈,𝑎_𝑦291=4𝑎_𝑦191−2𝑎_𝑥17−4𝑎₁₄𝑎₁₉+𝑎₂₁₇,𝜇₃=2𝑎^29𝑥𝑦1+𝑎_𝑦291𝑎₁₇−𝑎_𝑦291𝑎₂₈𝑎_𝑦1924,𝜇₄=2𝑎^19𝑥𝑦2−𝑎_𝑦192𝑎₁₇+𝑎_𝑦192𝑎₂₈𝜇₅4,𝜔₁=−𝜇₁+𝜇₂,𝜔₂=2𝑎_𝑦121−4𝑎_𝑦112−2𝑎₁₁𝑎₁₅+2𝑎₁₂𝑎₁₄−𝑎₁₂𝑎₂₅+4𝑎₁₃𝑎₂₄,𝜔₃=2𝑎_𝑦122−4𝑎_𝑦131+4𝑎₁₁𝑎₁₆−𝑎₁₂𝑎₁₅+2𝑎₁₂𝑎₂₆−2𝑎₁₃𝑎₂₅,=𝑎_𝑦151−2𝑎_𝑦142−𝑎_𝑦252+2𝑎_𝑦261−4𝑎₁₁𝑎₁₈+2𝑎₁₂𝑎₁₇𝜔₄−2𝑎₁₂𝑎₂₈+4𝑎₁₃𝑎₂₇,=2𝑎_𝑦152−4𝑎_𝑥13−4𝑎_𝑦161−2𝑎₁₃𝑎₁₇−2𝑎₁₃𝑎₂₈+4𝑎₁₄𝑎₁₆𝜔₅−𝑎₂₁₅+2𝑎₁₅𝑎₂₆−2𝑎₁₆𝑎₂₅,=4𝑎_𝑥12+4𝑎_𝑦252−8𝑎_𝑦261+8𝑎₁₁𝑎₁₈−2𝑎₁₂𝑎₁₇+6𝑎₁₂𝑎₂₈−8𝑎₁₃𝑎₂₇𝜔₆−2𝑎₁₅𝑎₂₅+8𝑎₁₆𝑎₂₄+𝜔₃,=4𝑎_𝑥11+4𝑎_𝑦242−2𝑎_𝑦251+2𝑎₁₁𝑎₁₇+2𝑎₁₁𝑎₂₈−2𝑎₁₄𝑎₂₅+2𝑎₁₅𝑎₂₄𝜔₇−4𝑎₂₄𝑎₂₆+𝑎₂₂₅,𝜔₈=2𝑎_𝑦182−4𝑎_𝑥16−4𝑎₁₃𝑎₁₉−𝑎₁₅𝑎₁₈+2𝑎₁₆𝑎₁₇−2𝑎₁₆𝑎₂₈+2𝑎₁₈𝑎₂₆,=5𝑎_𝑥15−𝑎_𝑦172−4𝑎_𝑦181−6𝑎_𝑥26+3𝑎_𝑦282+4𝑎₁₂𝑎₁₉−8𝑎₁₃𝑎₂₉𝜔₉+4𝑎₁₄𝑎₁₈−2𝑎₁₅𝑎₁₇+2𝑎₁₅𝑎₂₈+4𝑎₁₆𝑎₂₇−4𝑎₁₈𝑎₂₅,𝜔₁₀=𝑎_𝑦171−2𝑎_𝑥14+3𝑎_𝑥25−3𝑎_𝑦281+4𝑎₁₂𝑎₂₉−2𝑎₁₅𝑎₂₇+4𝑎₁₈𝑎₂₄,𝜔₁₁=2𝑎_𝑦271−4𝑎_𝑥24−4𝑎₁₁𝑎₂₉+2𝑎₁₄a₂₇−2𝑎₁₇𝑎₂₄+2𝑎₂₄𝑎₂₈−𝑎₂₅𝑎₂₇,=4𝑎_𝑥25+4𝑎_𝑦272−8𝑎_𝑦281+8𝑎₁₁𝑎₁₉+8𝑎₁₂𝑎₂₉−2𝑎₁₅𝑎₂₇−2𝑎₁₇𝑎₂₅𝜔₁₂+8𝑎₁₈𝑎₂₄+2𝑎₂₅𝑎₂₈−4𝑎₂₆𝑎₂₇,=12𝑎_𝑦172−12𝑎_𝑦181−8𝑎_𝑥26+4𝑎_𝑦282−8𝑎₁₂𝑎₁₉−24𝑎₁₃𝑎₂₉+12𝑎₁₄𝑎₁₈𝜔₁₃−6𝑎₁₅𝑎₁₇+6𝑎₁₅𝑎₂₈−8𝑎₁₆𝑎₂₇−2𝑎₁₈𝑎₂₅,=2𝜆₁₅,𝜔₁₄=4𝜆₁₆,𝜔₁₅=−2𝜆₁₂.(A.1)

Acknowledgments

This research was supported by the Royal Golden Jubilee Ph.D. Program of Thailand (TRF). The authors are also thankful to E. Schulz for fruitful discussions.