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

Authors: Sakka Sookmee; Sergey V. Meleshko
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

The necessary form of a linearizable system of two second-order ordinary differential equationsy1″=f1(x,y1,y2,y1′,y2′), y2″=f2(x,y1,y2,y1′,y2′) 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.


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## Body

## 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 ofn second-order ordinary differential equations is(1.1)v̈+Cv̇+Dv+E=0,
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 (1.2)ü+K1u̇=0
or(1.3)ü+Ku=0.
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:(1.4)K=(a00b),K=(a10a),K=(ac-ca),
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 whereK=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) is(1.5)ü1+k1u1+k3u2=0,ü2+k4u1+k2u2=0,
where ki, (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 equation(1.6)u1(4)+(k1+k2)ü1+(k1k2-k3k4)u1=0.
Here,(1.7)K=(k1k3k4k2).
The general solution of the last equation depends on the roots λ of the characteristic equation(1.8)λ4+(k1+k2)λ2+(k1k2-k3k4)=0.
The solution is similar for k4≠0. On the other hand, if k3=0 and k4=0, then system (1.1) is decoupled:(1.9)ü1+k1u1=0,ü2+k2u2=0.
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 equations(1.10)y1′′=F1(x,y1,y2,y1′,y2′),y2′′=F2(x,y1,y2,y1′,y2′).
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(1.11)t=φ(x,y1,y2),u1=ψ1(x,y1,y2),u2=ψ2(x,y1,y2)
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 transformation(1.12)t=φ(x),u1=ψ1(x,y1,y2),u2=ψ2(x,y1,y2)
to system (1.5).The paper is organized as follows. In Section2, 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.


## 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 ofn second-order ordinary differential equations is(1.1)v̈+Cv̇+Dv+E=0,
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 (1.2)ü+K1u̇=0
or(1.3)ü+Ku=0.
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:(1.4)K=(a00b),K=(a10a),K=(ac-ca),
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 whereK=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) is(1.5)ü1+k1u1+k3u2=0,ü2+k4u1+k2u2=0,
where ki, (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 equation(1.6)u1(4)+(k1+k2)ü1+(k1k2-k3k4)u1=0.
Here,(1.7)K=(k1k3k4k2).
The general solution of the last equation depends on the roots λ of the characteristic equation(1.8)λ4+(k1+k2)λ2+(k1k2-k3k4)=0.
The solution is similar for k4≠0. On the other hand, if k3=0 and k4=0, then system (1.1) is decoupled:(1.9)ü1+k1u1=0,ü2+k2u2=0.
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 equations(1.10)y1′′=F1(x,y1,y2,y1′,y2′),y2′′=F2(x,y1,y2,y1′,y2′).
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(1.11)t=φ(x,y1,y2),u1=ψ1(x,y1,y2),u2=ψ2(x,y1,y2)
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 transformation(1.12)t=φ(x),u1=ψ1(x,y1,y2),u2=ψ2(x,y1,y2)
to system (1.5).The paper is organized as follows. In Section2, 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
(2.1)u1′=g1(x,y1,y2,y1′,y2′)=Dxψ1Dxφ,u1′′=Dxg1Dxφ,u2′=g2(x,y1,y2,y1′,y2′)=Dxψ2Dxφ,u2′′=Dxg2Dxφ,
where(2.2)Dx=∂∂x+y1′∂∂y1+y2′∂∂y2+y1′′∂∂y1′+y2′′∂∂y2′.
Replacing u1′, u1′′, u2′, and u2′′ in system (1.3), it becomes(2.3)y1′′=y1′(a11y1′2+a12y1′y2′+a13y2′2)+a14y1′2+a15y1′y2′+a16y2′2+a17y1′+a18y2′+a19,y2′′=y2′(a11y1′2+a12y1′y2′+a13y2′2)+a24y1′2+a25y1′y2′+a26y2′2+a27y1′+a28y2′+a29,
where the coefficients aij are expressed through the functions φ, ψ1, ψ2, their partial derivatives, and the entries of the matrix K=(kij(t)) as follows:(2.4)a11=(h1ψ1y1y1+h2ψ2y1y1-vφy1y1+f1φy13-f2φy12φy2)Δ,a12=2(h1ψ1y1y2+h2ψ2y1y2-vφy1y2+f1φy12φy2-f2φy1φy22)Δ,a13=(h1ψ1y2y2+h2ψ2y2y2-vφy2y2+f1φy1φy22-f2φy23)Δ,a14=(2h1ψ1xy1+2h2ψ2xy1+h3ψ1y1y1+h4ψ2y1y1+h5φy1y1-2vφxy1+3f1φxφy12-2f2φxφy1φy2-J)Δ,a15=2(h1ψ1xy2+h2ψ2xy2+h3ψ1y1y2+h4ψ2y1y2+h5φy1y2-2vφxy2+2f1φxφy1φy2-f2φxφy22-R)Δ,a16=(h3ψ1y2y2+h4ψ2y2y2+h5φy2y2+f1φxφy22-f3φy23)Δ,a17=(h1ψ1xx+h2ψ2xx+2h3ψ1xy1+2h4ψ2xy1+2h5φxy1-vφxx+3f1φx2φy1-f2φx2φy2-2f3φxφy1)Δ,a18=2(h3ψ1xy2+h4ψ2xy2+2h5φxy2+f1φx2φy2-f3φxφy22)Δ,a19=(h3ψ1xx+h4ψ2xx+h5φxx+f1φx3-f3φx2φy2)Δ,a24=(h6ψ1y1y1+h7ψ2y1y1+h8φy1y1-f2φxφy12+f3φy13)Δ,a25=2(h1ψ1xy1+h2ψ2xy1-vφxy1+h6ψ1y1y2+h7ψ2y1y2+f1φxφxφy12-2f2φxφy1φy2-f3φy12φy2)Δ,a26=(2h1ψ1xy2+2h2ψ2xy2-2vφxy2+h6ψ1y2y2+h7ψ2y2y2+h8φy2y2+2f1φxφy1φy2-3f2φxφy22-R)Δ,a27=2(h6ψ1xy1+h7ψ2xy1+h8φxy1-f2φx2φy1+f3φxφy12)Δ,a28=(h1ψ1xx+h2ψ2xx-vφxx+2h6ψ1xy2+2h7ψ2xy2+2h8φxy2+f1φx2φy1-3f2φx2φy2+U)Δ,a29=(h6ψ1xx+h7ψ2xx+h8φxx-f2φx3+f3φx2φy1)Δ,
where 𝔍  denotes f3φy12φy2, ℜ denotes f3φy1φy22, 𝔘 denotes 2f3φxφy1φy2, and Δ≠0 is the Jacobian of the change of variables  (1.11)(2.5)Δ=(φxψ1y1ψ2y2-φxψ1y2ψ2y1-φy1ψ1xψ2y2+φy1ψ1y2ψ2x+φy2ψ1xψ2y1-φy2ψ1y1ψ2x),f1=ψ1y2(k22ψ2+k21ψ1)-ψ2y2(k11ψ1+k12ψ2),f2=ψ1y1(k22ψ2+k21ψ1)-ψ2y1(k11ψ1+k12ψ2),f3=ψ1x(k22ψ2+k21ψ1)-ψ2x(k11ψ1+k12ψ2),v=ψ1y2ψ2y1-ψ1y1ψ2y2,h1=φy2ψ2y1-φy1ψ2y2,h2=φy1ψ1y2-φy2ψ1y1,h3=φy2ψ2x-φxψ2y2,h4=φxψ1y2-φy2ψ1x,h5=ψ1xψ2y2-ψ1y2ψ2x,h6=φxψ2y1-φy1ψ2x,h7=φy1ψ1x-φxψ1y1,h8=ψ1y1ψ2x-ψ1xψ2y1.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 aij of system (2.3). The compatibility analysis depends on the value of ψ1y1.The next part of the present paper deals with a fiber preserving set of point transformations (1.12):(3.1)t=φ(x),u1=ψ1(x,y1,y2),u2=ψ2(x,y1,y2),
and constant matrix(3.2)K=(k1k3k4k2).


### 3.1. Caseψ1y1≠0

Substitution ofφy1=0 and φy2=0 into (2.4) gives
(3.3)a11=0,a12=0,a13=0,(3.4)vx=(2φxxv-φxv(a17+a28))(2φx),vy1=-v(2a14+a25)2,vy2=-v(a15+2a26)2,(3.5)ψ1xx=(φxxψ1x-φx3(k1ψ1+k3ψ2)-φxψ1y1a19-φxψ1y2a29)φx,(3.6)ψ1xy1=(φxxψ1y1-φxψ1y1a17-φxψ1y2a27)(2φx),(3.7)ψ1xy2=(φxxψ1y2-φxψ1y1a18-φxψ1y2a28)(2φx),(3.8)ψ1y1y1=-(ψ1y1a14+ψ1y2a24),(3.9)ψ1y1y2=-(ψ1y1a15+ψ1y2a25)2,(3.10)ψ1y2y2=-(ψ1y1a16+ψ1y2a26),(3.11)ψ2y2=(ψ1y2ψ2y1-v)ψ1y1,(3.12)ψ2y1y1=(a24v-ψ1y1ψ2y1a14-ψ1y2ψ2y1a24)ψ1y1,(3.13)ψ2xy1=(φxxψ1y1ψ2y1-φxψ1y1ψ2y1a17-φxψ1y2ψ2y1a27+φxa27v)(2φxψ1y1),(3.14)ψ2xx=(φxxψ1y1ψ2x-φx3ψ1y1(k2ψ2+k4ψ1)-φxψ1y1ψ2y1a19-φxψ1y2ψ2y1a29+φxa29v)(φxψ1y1),
where Δ=-φxv≠0. Comparing the mixed derivatives (vx)y1=(vy1)x,(vx)y2=(vy2)x and (vy2)y1=(vy1)y2, one obtains (3.15)2a14x-a17y1+a25x-a28y1=0,a15x-a17y2+2a26x-a28y2=0,a15y1-2a14y2-a25y2+2a26y1=0.
Considering the conditions (ψixx)yj=(ψixyj)x(i,j=1,2), one has
(3.16)φxxx=(3φxx2ψ1y1-4φx4ψ1y1k1-4φx4ψ2y1k3+φx2ψ1y1(λ20-λ16)+φx2ψ1y2λ12)(2φxψ1y1),(3.17)k4=(4φx2ψ1y1ψ2y1(k1-k2)+4φx2ψ2y12k3-vλ12)(4φx2ψ1y12),(3.18)k3=(ψ1y22λ12-ψ1y12λ15-ψ1y1ψ1y2λ16)(4φx2v),k2=(4φx2ψ1y1k1v-2ψ1y12ψ2y1λ15-2ψ1y1ψ1y2ψ2y1λ16+ψ1y1vλ16+2ψ1y22ψ2y1λ12-2ψ1y2vλ12)(4φx2ψ1y1v).
Here, the functions λn(x,y1,y2) are defined through aij(x,y1,y2) and their derivatives (presented in the appendix).Equating the mixed derivatives(ψixy1)y2=(ψixy2)y1, (ψixy1)y1=(ψiy1y1)x, (ψ1xy2)yi=(ψ1yiy2)x, (ψ1y1yi)y2=(ψ1yiy2)y1, (ψ2y1y1)y2=(ψ2y2)y1y1 and using the conditions (ki+1)y2=0, (i=1,2), one obtains(3.19)λn=0(n=1,2,…,11).
Note that (ψ1xy2)y1-(ψ1y1y2)x=0 is satisfied. Differentiating (3.16) and (3.17) with respect to y1 and y2, one has(3.20)2λ15y1-2a14λ15-a15λ16+a25λ15=0,λ27+j=0(j=0,1,2).Equations(ki)x=0(i=2,3,4), become
(3.21)4φxxλ12+φxλ14=0,(3.22)2φxxλ16-φx(λ16x+a18λ12-a27λ15)=0,(3.23)4φxxλ15+φx(a17λ15-2λ15x+a18λ16-a28λ15)=0.
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. Caseλ12≠0

Substitutingφxx, found from (3.21), into (3.23) and (3.22), one obtains(3.24)a17λ12λ15-2λ15xλ12+a18λ12λ16-a28λ12λ15-λ14λ15=0,2λ16xλ12+2a18λ122-2a27λ12λ15+λ14λ16=0.
Differentiating (3.21) with respect to y1 and y2, one has(3.25)λ12y1λ14-λ14y1λ12=0,λ12y2λ14-λ14y2λ12=0.
Equation (3.16) becomes(3.26)k1=(16ψ1y12ψ2y1λ122λ15+16ψ1y1ψ1y2ψ2y1λ122λ16+16ψ1y1λ122v(λ20-λ16)+4ψ1y1λ12v(2λ14x+a17λ14-a28λ14)+ψ1y1λ14v(4a27λ16+5λ14)-16ψ1y22ψ2y1λ123+16ψ1y2λ123v)/(64φx2ψ1y1λ122v).
Differentiating (3.26) with respect to x, one gets the condition(3.27)32λ123λ17+8λ122λ18+2λ12λ19+λ14(8a272λ162+18a27λ14λ16+15λ142)=0.
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. Caseλ12=0 and λ16≠0

Sinceλ12=0 and φx≠0, (3.21) leads to the condition λ14=0, and (3.22) becomes(3.28)φxx=φxλ16x(2λ16).
Substituting φxx into (3.23) and (3.16), one gets(3.29)2(λ16xλ15-λ15xλ16)+λ15λ16(a17-a28)+a18λ162=0,(3.30)k1=(4ψ1y1ψ2y1λ162λ15+4ψ1y2ψ2y1λ163-4λ163v+4λ162λ23v-4λ16λ16xxv+5λ16x2v)(16φx2λ162v).
Equation (k1)x=0 leads to the condition
(3.31)8λ163λ21+4λ162λ22+18λ16λ16xxλ16x-15λ16x3=0.
Note that (φxx)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. Caseλ12=0,λ16=0, and λ15≠0

Substitutingφxx, found from (3.23), into (3.16), one has(3.32)k1=(16ψ1y1ψ2y1λ153+λ26v)(64φx2λ152v).
Differentiating (3.32) with respect to x, one gets(3.33)λ153λ24+2λ152λ25-120λ15x3+36λ15λ15x(4λ15xx+λ15xa17-λ15xa28)=0.
Note that the equations (φxx)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 caseλ12=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 caseψ1y1≠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)
Ifλ12≠0, then the additional conditions are (3.24), (3.25), and (3.27).(I.2)
Ifλ12=0 and λ16≠0, then the additional conditions are (3.29) and (3.31).(I.3)
Ifλ12=0, λ16=0, and λ15≠0, then the additional condition is (3.33).(I.4)
Ifλ12=0, λ16=0, and λ15=0, then there are no additional conditions.


### 3.2. Caseψ1y1=0

Ifψ2y2≠0, then the change x¯=x,y1¯=y2,y2¯=y1 leads to the previous case. Hence, without loss of generality one can assume that ψ2y2=0. Equations (2.4) become(3.34)a11=0,a12=0,a13=0,a15=0,a16=0,a18=0,a24=0,a25=0,a27=0,(3.35)ψ2y1y1=-ψ2y1a14,ψ2xy1=(φxxψ2y1-φxψ2y1a17)(2φx),ψ1y2y2=-ψ1y2a26,ψ1xy2=(φxxψ1y2-φxψ1y2a28)(2φx),ψ1xx=(φxxψ1x-φx3(k1ψ1+k3ψ2)-φxψ1y2a29)φx,ψ2xx=(φxxψ2x-φx3(k2ψ2+k4ψ1)-φxψ2y1a19)φx,
and Δ=-φxψ1y2ψ2y1≠0.Comparing all mixed derivatives of third-order of the functionsψj, j=1,2, one finds(3.36)a28y1=0,a28y2=2a26x,a26y1=0,a17y2=0,a17y1=2a14x,a14y2=0,(3.37)k1=(3φxx2-2φxxxφx+φx2μ1)(4φx4),k2=(3φxx2-2φxxxφx-φx2μ2)(4φx4),k3=(-a29y1ψ1y2)(φx2ψ2y1),k4=(-a19y2ψ2y1)(φx2ψ1y2),
where the coefficients μn are defined through aij and their derivatives (presented in the appendix).Equations(ki)x=0, (ki)yj=0(i=1,2,3,4,j=1,2), become(3.38)φxxxx=(12φxxxφxxφx-12φxx3+2φxxφx2μ2-φx3μ2x)(2φx2),(3.39)(φxx-φxμ3)a29y1=0,(φxx-φxμ4)a19y2=0,2φxxμ5-φxμ5x=0,(3.40)a19y2a14-a19y1y2=0,a29y1a26-a29y1y2=0,a29y1y1+a29y1a14=0,a19y2y2+a19y2a26=0,a26xx-a26xa28+a26y2a29-a29y2y2+a29y2a26=0,a14xx-a14xa17+a14y1a19-a19y1y1+a19y1a14=0.
Notice that (φxxxx)yj=0(j=1,2) are satisfied. Equation (3.39) generates a further analysis of the compatibility depending on the values of a29y1, a19y2, and μ5.


#### 3.2.1. Casea29y1≠0

From the first equation of (3.39), one obtains that(3.41)φxx=φxμ3.
Substituting φxx into the second and third equations of (3.39), one obtains(3.42)2μ5μ3-μ5x=0,a19y2(μ3-μ4)=0.
Substitution of φxx into (3.38) gives(3.43)2μ3xx+μ2x-6μ3xμ3-2μ2μ3+2μ33=0.
Note that (φxx)y1=0 and (φxx)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. Casea29y1=0 and a19y2≠0

From the second equation of (3.39), one obtains that(3.44)φxx=φxμ4.
Substituting φxx into the third equation of (3.39), one obtains(3.45)μ5x-2μ5μ4=0.
Substitution of φxx into (3.38) gives(3.46)6μ4xμ4-μ2x-2μ4xx+2μ2μ4-2μ43=0.
Note that (φxx)y1=0 and (φxx)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. Casea29y1=0,a19y2=0, and μ5≠0

From the third equation of (3.39), one obtains(3.47)φxx=(φxμ5x)(2μ5).
Substitution of φxx into (3.38) leads to the condition(3.48)μ52(4μ5xμ2-4μ5xxx)+18μ5xxμ5xμ5-15μ5x3-4μ2xμ53=0.
Note that (φxx)y1=0 and (φxx)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 that(3.49)k2=k1+μ5(4φx2),k3=0,k4=0.Remark 3.3.
In the casea29y1=0,a19y2=0,μ5=0, one has k2=k1 and k3=k4=0. This case corresponds to (1.9).Combining all obtained results in the caseψ1y1=0 and ψ2y2=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)
Ifa29y1≠0, then the additional conditions are (3.42) and (3.43).(II.2)
Ifa29y1=0 and a19y2≠0, then the additional conditions are (3.45) and (3.46).(II.3)
Ifa29y1=0,a19y2=0, and μ5≠0, then the additional condition is (3.48).(II.4)
Ifa29y1=0,a19y2=0, and μ5=0, then there are no additional conditions.Remark 3.5.
If one assumes that the conditions (II) of Theorem3.4 are valid, then the conditions (I) of Theorem 3.2 vanish. Moreover, these conditions also imply that λ12=-4a29y1, λ15=-4a19y2, 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 ψ1y1 and ψ2y2.
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].


## 3.1. Caseψ1y1≠0

Substitution ofφy1=0 and φy2=0 into (2.4) gives
(3.3)a11=0,a12=0,a13=0,(3.4)vx=(2φxxv-φxv(a17+a28))(2φx),vy1=-v(2a14+a25)2,vy2=-v(a15+2a26)2,(3.5)ψ1xx=(φxxψ1x-φx3(k1ψ1+k3ψ2)-φxψ1y1a19-φxψ1y2a29)φx,(3.6)ψ1xy1=(φxxψ1y1-φxψ1y1a17-φxψ1y2a27)(2φx),(3.7)ψ1xy2=(φxxψ1y2-φxψ1y1a18-φxψ1y2a28)(2φx),(3.8)ψ1y1y1=-(ψ1y1a14+ψ1y2a24),(3.9)ψ1y1y2=-(ψ1y1a15+ψ1y2a25)2,(3.10)ψ1y2y2=-(ψ1y1a16+ψ1y2a26),(3.11)ψ2y2=(ψ1y2ψ2y1-v)ψ1y1,(3.12)ψ2y1y1=(a24v-ψ1y1ψ2y1a14-ψ1y2ψ2y1a24)ψ1y1,(3.13)ψ2xy1=(φxxψ1y1ψ2y1-φxψ1y1ψ2y1a17-φxψ1y2ψ2y1a27+φxa27v)(2φxψ1y1),(3.14)ψ2xx=(φxxψ1y1ψ2x-φx3ψ1y1(k2ψ2+k4ψ1)-φxψ1y1ψ2y1a19-φxψ1y2ψ2y1a29+φxa29v)(φxψ1y1),
where Δ=-φxv≠0. Comparing the mixed derivatives (vx)y1=(vy1)x,(vx)y2=(vy2)x and (vy2)y1=(vy1)y2, one obtains (3.15)2a14x-a17y1+a25x-a28y1=0,a15x-a17y2+2a26x-a28y2=0,a15y1-2a14y2-a25y2+2a26y1=0.
Considering the conditions (ψixx)yj=(ψixyj)x(i,j=1,2), one has
(3.16)φxxx=(3φxx2ψ1y1-4φx4ψ1y1k1-4φx4ψ2y1k3+φx2ψ1y1(λ20-λ16)+φx2ψ1y2λ12)(2φxψ1y1),(3.17)k4=(4φx2ψ1y1ψ2y1(k1-k2)+4φx2ψ2y12k3-vλ12)(4φx2ψ1y12),(3.18)k3=(ψ1y22λ12-ψ1y12λ15-ψ1y1ψ1y2λ16)(4φx2v),k2=(4φx2ψ1y1k1v-2ψ1y12ψ2y1λ15-2ψ1y1ψ1y2ψ2y1λ16+ψ1y1vλ16+2ψ1y22ψ2y1λ12-2ψ1y2vλ12)(4φx2ψ1y1v).
Here, the functions λn(x,y1,y2) are defined through aij(x,y1,y2) and their derivatives (presented in the appendix).Equating the mixed derivatives(ψixy1)y2=(ψixy2)y1, (ψixy1)y1=(ψiy1y1)x, (ψ1xy2)yi=(ψ1yiy2)x, (ψ1y1yi)y2=(ψ1yiy2)y1, (ψ2y1y1)y2=(ψ2y2)y1y1 and using the conditions (ki+1)y2=0, (i=1,2), one obtains(3.19)λn=0(n=1,2,…,11).
Note that (ψ1xy2)y1-(ψ1y1y2)x=0 is satisfied. Differentiating (3.16) and (3.17) with respect to y1 and y2, one has(3.20)2λ15y1-2a14λ15-a15λ16+a25λ15=0,λ27+j=0(j=0,1,2).Equations(ki)x=0(i=2,3,4), become
(3.21)4φxxλ12+φxλ14=0,(3.22)2φxxλ16-φx(λ16x+a18λ12-a27λ15)=0,(3.23)4φxxλ15+φx(a17λ15-2λ15x+a18λ16-a28λ15)=0.
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. Caseλ12≠0

Substitutingφxx, found from (3.21), into (3.23) and (3.22), one obtains(3.24)a17λ12λ15-2λ15xλ12+a18λ12λ16-a28λ12λ15-λ14λ15=0,2λ16xλ12+2a18λ122-2a27λ12λ15+λ14λ16=0.
Differentiating (3.21) with respect to y1 and y2, one has(3.25)λ12y1λ14-λ14y1λ12=0,λ12y2λ14-λ14y2λ12=0.
Equation (3.16) becomes(3.26)k1=(16ψ1y12ψ2y1λ122λ15+16ψ1y1ψ1y2ψ2y1λ122λ16+16ψ1y1λ122v(λ20-λ16)+4ψ1y1λ12v(2λ14x+a17λ14-a28λ14)+ψ1y1λ14v(4a27λ16+5λ14)-16ψ1y22ψ2y1λ123+16ψ1y2λ123v)/(64φx2ψ1y1λ122v).
Differentiating (3.26) with respect to x, one gets the condition(3.27)32λ123λ17+8λ122λ18+2λ12λ19+λ14(8a272λ162+18a27λ14λ16+15λ142)=0.
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. Caseλ12=0 and λ16≠0

Sinceλ12=0 and φx≠0, (3.21) leads to the condition λ14=0, and (3.22) becomes(3.28)φxx=φxλ16x(2λ16).
Substituting φxx into (3.23) and (3.16), one gets(3.29)2(λ16xλ15-λ15xλ16)+λ15λ16(a17-a28)+a18λ162=0,(3.30)k1=(4ψ1y1ψ2y1λ162λ15+4ψ1y2ψ2y1λ163-4λ163v+4λ162λ23v-4λ16λ16xxv+5λ16x2v)(16φx2λ162v).
Equation (k1)x=0 leads to the condition
(3.31)8λ163λ21+4λ162λ22+18λ16λ16xxλ16x-15λ16x3=0.
Note that (φxx)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. Caseλ12=0,λ16=0, and λ15≠0

Substitutingφxx, found from (3.23), into (3.16), one has(3.32)k1=(16ψ1y1ψ2y1λ153+λ26v)(64φx2λ152v).
Differentiating (3.32) with respect to x, one gets(3.33)λ153λ24+2λ152λ25-120λ15x3+36λ15λ15x(4λ15xx+λ15xa17-λ15xa28)=0.
Note that the equations (φxx)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 caseλ12=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 caseψ1y1≠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)
Ifλ12≠0, then the additional conditions are (3.24), (3.25), and (3.27).(I.2)
Ifλ12=0 and λ16≠0, then the additional conditions are (3.29) and (3.31).(I.3)
Ifλ12=0, λ16=0, and λ15≠0, then the additional condition is (3.33).(I.4)
Ifλ12=0, λ16=0, and λ15=0, then there are no additional conditions.


## 3.1.1. Caseλ12≠0

Substitutingφxx, found from (3.21), into (3.23) and (3.22), one obtains(3.24)a17λ12λ15-2λ15xλ12+a18λ12λ16-a28λ12λ15-λ14λ15=0,2λ16xλ12+2a18λ122-2a27λ12λ15+λ14λ16=0.
Differentiating (3.21) with respect to y1 and y2, one has(3.25)λ12y1λ14-λ14y1λ12=0,λ12y2λ14-λ14y2λ12=0.
Equation (3.16) becomes(3.26)k1=(16ψ1y12ψ2y1λ122λ15+16ψ1y1ψ1y2ψ2y1λ122λ16+16ψ1y1λ122v(λ20-λ16)+4ψ1y1λ12v(2λ14x+a17λ14-a28λ14)+ψ1y1λ14v(4a27λ16+5λ14)-16ψ1y22ψ2y1λ123+16ψ1y2λ123v)/(64φx2ψ1y1λ122v).
Differentiating (3.26) with respect to x, one gets the condition(3.27)32λ123λ17+8λ122λ18+2λ12λ19+λ14(8a272λ162+18a27λ14λ16+15λ142)=0.
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. Caseλ12=0 and λ16≠0

Sinceλ12=0 and φx≠0, (3.21) leads to the condition λ14=0, and (3.22) becomes(3.28)φxx=φxλ16x(2λ16).
Substituting φxx into (3.23) and (3.16), one gets(3.29)2(λ16xλ15-λ15xλ16)+λ15λ16(a17-a28)+a18λ162=0,(3.30)k1=(4ψ1y1ψ2y1λ162λ15+4ψ1y2ψ2y1λ163-4λ163v+4λ162λ23v-4λ16λ16xxv+5λ16x2v)(16φx2λ162v).
Equation (k1)x=0 leads to the condition
(3.31)8λ163λ21+4λ162λ22+18λ16λ16xxλ16x-15λ16x3=0.
Note that (φxx)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. Caseλ12=0,λ16=0, and λ15≠0

Substitutingφxx, found from (3.23), into (3.16), one has(3.32)k1=(16ψ1y1ψ2y1λ153+λ26v)(64φx2λ152v).
Differentiating (3.32) with respect to x, one gets(3.33)λ153λ24+2λ152λ25-120λ15x3+36λ15λ15x(4λ15xx+λ15xa17-λ15xa28)=0.
Note that the equations (φxx)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 caseλ12=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 caseψ1y1≠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)
Ifλ12≠0, then the additional conditions are (3.24), (3.25), and (3.27).(I.2)
Ifλ12=0 and λ16≠0, then the additional conditions are (3.29) and (3.31).(I.3)
Ifλ12=0, λ16=0, and λ15≠0, then the additional condition is (3.33).(I.4)
Ifλ12=0, λ16=0, and λ15=0, then there are no additional conditions.


## 3.2. Caseψ1y1=0

Ifψ2y2≠0, then the change x¯=x,y1¯=y2,y2¯=y1 leads to the previous case. Hence, without loss of generality one can assume that ψ2y2=0. Equations (2.4) become(3.34)a11=0,a12=0,a13=0,a15=0,a16=0,a18=0,a24=0,a25=0,a27=0,(3.35)ψ2y1y1=-ψ2y1a14,ψ2xy1=(φxxψ2y1-φxψ2y1a17)(2φx),ψ1y2y2=-ψ1y2a26,ψ1xy2=(φxxψ1y2-φxψ1y2a28)(2φx),ψ1xx=(φxxψ1x-φx3(k1ψ1+k3ψ2)-φxψ1y2a29)φx,ψ2xx=(φxxψ2x-φx3(k2ψ2+k4ψ1)-φxψ2y1a19)φx,
and Δ=-φxψ1y2ψ2y1≠0.Comparing all mixed derivatives of third-order of the functionsψj, j=1,2, one finds(3.36)a28y1=0,a28y2=2a26x,a26y1=0,a17y2=0,a17y1=2a14x,a14y2=0,(3.37)k1=(3φxx2-2φxxxφx+φx2μ1)(4φx4),k2=(3φxx2-2φxxxφx-φx2μ2)(4φx4),k3=(-a29y1ψ1y2)(φx2ψ2y1),k4=(-a19y2ψ2y1)(φx2ψ1y2),
where the coefficients μn are defined through aij and their derivatives (presented in the appendix).Equations(ki)x=0, (ki)yj=0(i=1,2,3,4,j=1,2), become(3.38)φxxxx=(12φxxxφxxφx-12φxx3+2φxxφx2μ2-φx3μ2x)(2φx2),(3.39)(φxx-φxμ3)a29y1=0,(φxx-φxμ4)a19y2=0,2φxxμ5-φxμ5x=0,(3.40)a19y2a14-a19y1y2=0,a29y1a26-a29y1y2=0,a29y1y1+a29y1a14=0,a19y2y2+a19y2a26=0,a26xx-a26xa28+a26y2a29-a29y2y2+a29y2a26=0,a14xx-a14xa17+a14y1a19-a19y1y1+a19y1a14=0.
Notice that (φxxxx)yj=0(j=1,2) are satisfied. Equation (3.39) generates a further analysis of the compatibility depending on the values of a29y1, a19y2, and μ5.


### 3.2.1. Casea29y1≠0

From the first equation of (3.39), one obtains that(3.41)φxx=φxμ3.
Substituting φxx into the second and third equations of (3.39), one obtains(3.42)2μ5μ3-μ5x=0,a19y2(μ3-μ4)=0.
Substitution of φxx into (3.38) gives(3.43)2μ3xx+μ2x-6μ3xμ3-2μ2μ3+2μ33=0.
Note that (φxx)y1=0 and (φxx)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. Casea29y1=0 and a19y2≠0

From the second equation of (3.39), one obtains that(3.44)φxx=φxμ4.
Substituting φxx into the third equation of (3.39), one obtains(3.45)μ5x-2μ5μ4=0.
Substitution of φxx into (3.38) gives(3.46)6μ4xμ4-μ2x-2μ4xx+2μ2μ4-2μ43=0.
Note that (φxx)y1=0 and (φxx)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. Casea29y1=0,a19y2=0, and μ5≠0

From the third equation of (3.39), one obtains(3.47)φxx=(φxμ5x)(2μ5).
Substitution of φxx into (3.38) leads to the condition(3.48)μ52(4μ5xμ2-4μ5xxx)+18μ5xxμ5xμ5-15μ5x3-4μ2xμ53=0.
Note that (φxx)y1=0 and (φxx)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 that(3.49)k2=k1+μ5(4φx2),k3=0,k4=0.Remark 3.3.
In the casea29y1=0,a19y2=0,μ5=0, one has k2=k1 and k3=k4=0. This case corresponds to (1.9).Combining all obtained results in the caseψ1y1=0 and ψ2y2=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)
Ifa29y1≠0, then the additional conditions are (3.42) and (3.43).(II.2)
Ifa29y1=0 and a19y2≠0, then the additional conditions are (3.45) and (3.46).(II.3)
Ifa29y1=0,a19y2=0, and μ5≠0, then the additional condition is (3.48).(II.4)
Ifa29y1=0,a19y2=0, and μ5=0, then there are no additional conditions.Remark 3.5.
If one assumes that the conditions (II) of Theorem3.4 are valid, then the conditions (I) of Theorem 3.2 vanish. Moreover, these conditions also imply that λ12=-4a29y1, λ15=-4a19y2, 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 ψ1y1 and ψ2y2.
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].


## 3.2.1. Casea29y1≠0

From the first equation of (3.39), one obtains that(3.41)φxx=φxμ3.
Substituting φxx into the second and third equations of (3.39), one obtains(3.42)2μ5μ3-μ5x=0,a19y2(μ3-μ4)=0.
Substitution of φxx into (3.38) gives(3.43)2μ3xx+μ2x-6μ3xμ3-2μ2μ3+2μ33=0.
Note that (φxx)y1=0 and (φxx)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. Casea29y1=0 and a19y2≠0

From the second equation of (3.39), one obtains that(3.44)φxx=φxμ4.
Substituting φxx into the third equation of (3.39), one obtains(3.45)μ5x-2μ5μ4=0.
Substitution of φxx into (3.38) gives(3.46)6μ4xμ4-μ2x-2μ4xx+2μ2μ4-2μ43=0.
Note that (φxx)y1=0 and (φxx)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. Casea29y1=0,a19y2=0, and μ5≠0

From the third equation of (3.39), one obtains(3.47)φxx=(φxμ5x)(2μ5).
Substitution of φxx into (3.38) leads to the condition(3.48)μ52(4μ5xμ2-4μ5xxx)+18μ5xxμ5xμ5-15μ5x3-4μ2xμ53=0.
Note that (φxx)y1=0 and (φxx)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 that(3.49)k2=k1+μ5(4φx2),k3=0,k4=0.Remark 3.3.
In the casea29y1=0,a19y2=0,μ5=0, one has k2=k1 and k3=k4=0. This case corresponds to (1.9).Combining all obtained results in the caseψ1y1=0 and ψ2y2=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)
Ifa29y1≠0, then the additional conditions are (3.42) and (3.43).(II.2)
Ifa29y1=0 and a19y2≠0, then the additional conditions are (3.45) and (3.46).(II.3)
Ifa29y1=0,a19y2=0, and μ5≠0, then the additional condition is (3.48).(II.4)
Ifa29y1=0,a19y2=0, and μ5=0, then there are no additional conditions.Remark 3.5.
If one assumes that the conditions (II) of Theorem3.4 are valid, then the conditions (I) of Theorem 3.2 vanish. Moreover, these conditions also imply that λ12=-4a29y1, λ15=-4a19y2, 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 ψ1y1 and ψ2y2.
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 ψ1y1≠0, from (2.4), one obtains the derivatives vx, vyj, φxx, φxyj, φyjyk, ψ1xx, ψlxyj, ψlyjyk, (j,k,l=1,2). Comparing the mixed derivatives of the functions v, φ, ψ1, and ψ2, one can find the expressions of the quantities(4.1)ωn(n=1,2,…,15),
where ωn are expressed through aij and their derivatives (shown in the appendix). Excluding the functions v, φ, ψ1, and ψ2 from these expressions, one obtains the conditions(4.2)Ji=0(i=1,2,…,15),
where(4.3)J1=ω1ω11-2ω1ω9+2ω10ω2-ω3ω6,J2=ω1ω5+2ω2ω6,J3=6ω1ω8-2ω1ω12+10ω11ω2-20ω2ω9-5ω32,J4=2ω10ω2-ω1ω9,J5=10ω1ω7+ω12ω2-3ω2ω8,J6=ω1ω8+ω11ω2-3ω2ω9,J7=4ω1ω12-2ω1ω8+10ω32+5ω3ω5,J8=10ω1ω13+ω12ω3-3ω3ω8,J9=ω1ω15-ω10ω3,J10=ω1ω14+ω11ω3-3ω3ω9,J11=2ω12ω2-ω2ω8+5ω3ω4,J12=ω13ω2-ω3ω7,J13=2ω15ω2-ω3ω9,J14=ω14ω2-ω3ω8,J15=2ω1ω4-2ω2ω3-ω2ω5.
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(5.1)y1′′=a(y1,y2)y1′2+2b(y1,y2)y1′y2′+c(y1,y2)y2′2,y2′′=d(y1,y2)y1′2+2e(y1,y2)y1′y2′+f(y1,y2)y2′2.
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
(5.2)Si=0(i=1,2,3,4),
where
(5.3)S1=ay2-by1+be-cd,S2=by2-cy1+(ac-b2)+(bf-ce),S3=dy2-ey1-(ae-bd)-(df-e2),S4=by1+fy1-ay2-ey2.
Application of fiber preserving transformation to system (5.1) also leads to the same conditions (5.2).Example 5.2.
Consider a nonlinear system(5.4)y1′′=-y1′2-y2′2-q1,y2′′=q2-2y1′y2′,
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),
(5.5)λ12=-4q2,λ14=0,λ15=-4q2,λ16=0.
Since q2≠0, then λ12≠0, and (3.21) becomes φxx=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:
(5.6)ü1+k1u1=0,ü2+k2u2=0,
and the linearizing transformation is
(5.7)t=x,u1=12e(y1-y2),u2=12e(y1+y2).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(5.8)y1′′=g1(x)y1+g2(x)y2,y2′′=g3(x)y2+g4(x)y1.
For system (5.8),
(5.9)λ12=-4g4,λ16=4(g1-g3),λ15=-4g2,λ21=-2g3x,λ22=4g3λ16x-λ16xxx.
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
(5.10)8λ163λ21+4λ162λ22+18λ16λ16xxλ16x-15λ16x3=0,
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 equationsy1′′=f1(x,y1,y2,y1′,y2′), y2′′=f2(x,y1,y2,y1′,y2′) 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.



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*Source: 452689-2011-10-31.xml*