Royal Society Publishing

On the complete integrability and linearization of nonlinear ordinary differential equations. V. Linearization of coupled second-order equations

V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan

Abstract

Linearization of coupled second-order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper, we describe a simple and straightforward method to derive linearizing transformations for a class of two coupled SNODEs. Our procedure gives several new types of linearizing transformations of both invertible and non-invertible kinds. In both cases, we provide algorithms to derive the general solution of the given SNODE. We illustrate the theory with potentially important examples.

1. Introduction

Continuing our study on the integrability and linearization of coupled second-order nonlinear ordinary differential equations (SNODEs), in this paper, we focus our attention on the linearization of two coupled SNODEs. This study arises not only for the completeness of part IV (Chandrasekar et al. 2009), but also to show the importance of unfinished tasks that exist in the theory of linearization of two coupled SNODEs. As far as the first point is concerned, we show that one can also solve a class of coupled SNODEs by transforming them into two second-order free particle equations and, from the solutions of the latter, one can construct the solution of the former, even though this is a non-trivial problem in many situations (one can also transform coupled nonlinear ODEs into uncoupled nonlinear ones, which has already been pointed out by us in the previous paper, i.e. part IV). Regarding the second point, we wish to stress the fact that linearization of coupled nonlinear ODEs is a vast area of research that is still in its early stage. In this paper, we show that, in spite of the difficulties which exist in this topic, one can make useful progress on certain issues, namely: (i) developing a method to deduce all linearizing transformations wherein the new dependent variables are functions of only the old dependent and independent variables and not derivatives of the dependent variables and (ii) developing a method of constructing solutions of nonlinear ODEs from the linear ones in the case of non-point transformations.

Even though the modern theory of linearization of nonlinear ODEs had originated and developed with the works of Lie, Tresse and Cartan (Mahomed & Leach 1989; Steeb 1993; Olver 1995; Ibragimov 1999; Chandrasekar et al. 2005), the entire subject was lying dormant for more than a century. Only recently, during the past two decades or so, has notable progress been made to linearize nonlinear ODEs through non-point (Duarte et al. 1994) or generalized transformations (Chandrasekar et al. 2006). For example, focussing our attention on single second-order ODEs, generalized Sundmann (Euler et al. 2003; Euler & Euler 2004) and generalized linearizing transformations (Chandrasekar et al. 2006) have been introduced to linearize a class of equations that cannot be linearized by invertible point transformations. As far as two coupled SNODEs are concerned, to our knowledge, most of the studies were focussed only on invertible point transformations, irrespective of whether it is an analytical approach or a geometrical formulation. For a survey on this topic, one may refer to the recent papers of Merker (2006) and Mahomed & Qadir (2007) and, for the earlier works in this direction, we cite Crampin et al. (1996), Fels (1995), Grossman (2000), Soh & Mahomed (2001) and Qadir (2007). More recently, Sookmee & Meleshko (2008) proposed a new algorithm to linearize the coupled second-order ODEs by sequentially reducing the order of the equation.

In this work, we aim to give a new dimension to the theoretical development of linearization of nonlinear dynamical systems having two degrees of freedom by proving that one can unearth a wide class of linearizing transformations besides invertible point transformations. Of course, the latter ones form a subclass of the new ones that we construct in this paper. In this study, we not only derive several new types of linearizing transformations, but also propose systematic procedures to derive the general solution in all these cases. We also wish to emphasize here that we derive all these transformations from the first two integrals alone, and thereby establish a potentially simple, straightforward and powerful approach in the theory of differential equations.

The plan of the paper is as follows. In §2a, we briefly describe the method of deriving linearizing transformations for a system of two coupled second-order ODEs. We show that one can have two classes of linearizing transformations, depending on the nature of the independent variables. If the new independent variables are the same (z1=z2), we put them in class A category, and if they are different (z1z2), then we put them in class B category. In §2b(i), we consider class A category and identify three types of linearizing transformations. In §2b(ii), we consider class B category and identify six types of linearizing transformations. In §3, we consider one specific example for each of the nine types of linearizing transformations we have identified and obtain general solutions to each one of them to demonstrate our procedure. Finally, we present our conclusions in §4.

2. Linearizing transformations

(a) Method of deriving linearizing transformations

To begin with, let us consider a system of two coupled SNODEs, R{t,x} (eqn (2.1) in part IV (Chandrasekar et al. 2009)), Embedded Image 2.1 Any transformation of the form T{t,x}, defined by Embedded Image 2.2 which transforms the given set of nonlinear ODEs (2.1) to the free particle equations Embedded Image 2.3 is called a linearizing transformation in the present work.

Let Embedded Image 2.4 be the first two integrals of motion of the coupled system (2.1) and that they can be explicitly found, if they exist, e.g. by using the generalized modified Prelle–Singer (PS) method formulated in part IV (Chandrasekar et al. 2009). Then, the following theorem ensures that the transformation can be deduced from Ii, i=1,2.

Theorem 2.1

Suppose a given nonlinear system R{t,x} of ODEs (2.1) is linearizable to a system of two uncoupled free particle equations through the linearizing transformation T{t,x} of the form (2.2), then the latter can be deduced from the first integrals

Embedded Image, i=1,2.

Proof.

Let us re-express each of the functions Embedded Image and Embedded Image in equation (2.4) as a product of two new functions, i.e. Embedded Image 2.5 Again, rewriting f3 and f4 as total time derivatives of another set of functions, say z1 and z2, respectively, i.e. dEmbedded Image and Embedded Image, equation (2.5) can be further recast as Embedded Image 2.6

Now identifying the functions f1(t,x,y)=w1 and f2(t,x,y)=w2 as the new dependent variables, equation (2.6) can be further recast in the form Embedded Image 2.7 where Embedded Image and Embedded Image are the redefined constants. Obviously, equation (2.3) follows straightforwardly from equation (2.7). Consequently, the new variables, zi and wi, i=1,2, defined by equation (2.2) help us to transform the given set of coupled SNODEs into two linear second-order ODEs that, in turn, lead to the solution by trivial integration. The variables wi and zi, i=1,2, then define the linearizing transformations for the given equation (2.1). ▪

It may be noted that, in general, the new dependent variables, w1 and w2, may also involve Embedded Image and Embedded Image, i.e. Embedded Image and Embedded Image, and this possibility may lead us to identify more generalized transformations such as point-contact and generalized-contact transformations. However, in this paper, we will confine ourselves only to the forms of w1 and w2 given by equation (2.2).

(b) The nature of transformations

An important question that we will focus upon in this paper is what are the possible forms of linearizing transformations one can unearth through the above procedure. We recall here that, in the case of scalar SNODEs, one has point generalized Sundman and generalized linearizing transformations (Chandrasekar et al. 2005, 2006). As far as the coupled SNODEs (2.1) are concerned, as there are two independent variables z1 and z2 as given in equation (2.2), one can choose them to be either the same, z1=z2 (class A), or different, z1z2 (class B). In the case of class A transformations, one can construct three different types of linearizing transformations, whereas for class B, one can formulate six different types of linearizing transformations, as we point out below. However, we also note that even further types of local transformations involving the variables Embedded Image and Embedded Image are possible, but these are not included in the present study.

(i) Class A linearizing transformations (z1=z2=z)

In the case of class A transformations, we have w1=f1(t,x,y),w2=f2(t,x,y), Embedded ImageEmbedded Image. Now appropriately restricting the form of f3 (=f4), one can identify three different types of linearizing transformations.

  1. Suppose z1=z2=z is a perfect differential function and wi, i=1,2, and z do not contain the variables Embedded Image and Embedded Image, then we call the resultant transformation, namely w1=f1(t,x,y), w2=f2(t,x,y) and z=f3(t,x,y), a point transformation of type I.

  2. On the other hand, if z is not a perfect differential function, and wi, i=1,2, and z do not contain the variables Embedded Image and Embedded Image, then we call the resultant transformation, namely w1=f1(t,x,y), w2=f2(t,x,y) and Embedded Image, a generalized Sundman transformation of type I.

  3. As a more general case, if we consider the independent variable z to contain the derivative terms also, i.e. w1=f1(t,x,y), w2=f2(t,x,y) and Embedded Image, then we call the resultant transformation a generalized linearizing transformation of type I.

In our analysis, we do not consider the possibility w1=f1(t,x,y), w2=f2(t,x,y) and Embedded Image because the procedure to handle it is different from the presently discussed linearizing transformations. This possibility will be studied separately.

(ii) Class B linearizing transformations (z1z2)

In the class B type of linearizing transformations, we have Embedded Image and Embedded Image, z1z2. Now appropriately restricting the forms of f3 and f4, one can obtain six different types of linearizing transformations.

  1. If z1 and z2 are perfect differential functions and wi and zi, i=1,2, do not contain the variables Embedded Image and Embedded Image, then we call the resultant transformation, namely w1=f1(t,x,y), w2=f2(t,x,y), z1=f3(t,x,y) and z2=f4(t,x,y), a point transformation of type II.

  2. Suppose z1 is a perfect differential function and z2 is not a perfect differential function or vice versa, and if z1 and z2 do not contain the variables Embedded Image and Embedded Image, then we call the resultant transformation, namely w1=f1(t,x,y), w2=f2(t,x,y), z1=f3(t,x,y) and Embedded Image or Embedded Image and z2=f4(t,x,y), a mixed point-generalized Sundman transformation.

  3. On the other hand, if any one of the independent variables contains the variables Embedded Image and Embedded Image, then we call the resultant transformation, namely w1=f1(t,x,y), w2=f2(t,x,y),z1=f3(t,x,y) and Embedded Image or Embedded Image and z2=f4(t,x,y), a mixed point-generalized linearizing transformation.

  4. Suppose the independent variables are not perfect differential functions and are also not functions of Embedded Image and Embedded Image, i.e. w1=f1(t,x,y), Embedded Image and Embedded Image, then we call the resultant transformation a generalized Sundman transformation of type II (GST II).

  5. Further, if one of the independent variables, say z1, does not contain the derivative terms, whereas the other independent variable z2 does contain the derivative terms or vice versa, i.e. w1=f1(t,x,y), w2=f2(t,x,y), Embedded Image and Embedded Image or Embedded Image and Embedded Image, then we call the resultant transformation a mixed generalized Sundman-generalized linearizing transformation.

  6. As a general case, if we allow both the independent variables z1 and z2 to be non-perfect differential functions and also to contain derivative terms, i.e. w1=f1(t,x,y), w2=f2(t,x,y), Embedded Image and Embedded Image, then the resultant transformation will be termed a generalized linearizing transformation of type II.

Finally, the possibility that w1=f1(t,x,y), w2=f2(t,x,y), Embedded Image and Embedded Image is not considered in this study and will be pursued separately.

3. Applications

In this section, we consider specific examples and illustrate each one of the linearizing transformations identified in the previous section so as to make clear the applicability of them under different situations.

(a) Class-A linearizing transformations (z1=z2)

(i) Example 1: point transformation of type I

Let us consider the system of SNODEs Embedded Image 3.1 where ω is an arbitrary constant. The first two integrals associated with equation (3.1), which can be obtained using the formulation given in §2 of part IV (Chandrasekar et al. 2009), can be written as Embedded Image 3.2 Rewriting equation (3.2) in the form of equations (2.5) and (2.6), we obtain Embedded Image 3.3 so that we can identify point transformation of type I as Embedded Image 3.4 Using the transformation (3.4), one can transform equation (3.1) to a set of free particle equations, namely d2w1/dz2=0 and d2w2/dz2=0, so that w1=I1z+I3 and w2=I2z+I4, where I3 and I4 are the integration constants. Substituting the expressions for wi, i=1,2, and z in the free particle solutions and rewriting the resultant expressions in terms of the old variables x and y, one obtains the general solution for equation (3.1) in the form Embedded Image 3.5 where Embedded Image and Embedded Image. Here, we point out that the nonlinear system (3.1) admits amplitude-independent frequency of oscillations.

In the above example, we have considered the new dependent variables w1 and w2 and independent variable z to be functions of only x,y and t. We will now consider examples that admit more general transformations.

(ii) Example 2: generalized Sundman transformation of type I

Let us focus our attention on the two-dimensional Mathews–Lakshmanan oscillator system of the form (Cariñena et al. 2004; Chandrasekar et al. 2009) Embedded Image 3.6 where Embedded Image and λ and α are arbitrary parameters. For α=0, equation (3.6) admits the following two integrals of motion: Embedded Image3.7 We note that the integrals I2 and I3 (eqns (5.35) and (5.36) in part IV; Chandrasekar et al. 2009) can be derived from equation (3.7) by using the relations Embedded Image and Embedded Image, where I1 is given in eqn (5.35) of part IV. The general case (α≠0) can be linearized through mixed generalized Sundman-generalized linearized transformation (example 8). To demonstrate the linearization through generalized Sundman transformation of type I, we here consider equation (3.6) with α=0.

For the case α=0, one may note that, on making a substitution y(t) = y(x(t)) into equation (3.6), one can obtain a non-autonomous second-order ODE in y(x). Although this equation satisfies the linearization condition for point transformation (Sookmee & Meleshko 2008), finding the linearizing transformation and the general solution for the transformed ODE turns out to be non-trivial. On the other hand, we provide a straightforward procedure of linearization.

The above two integrals (3.7) can be rewritten as Embedded Image 3.8 and Embedded Image 3.9 From the above equations, we identify the new dependent and independent variables as Embedded Image 3.10 One may observe that the independent variables z1 and z2 are not perfect differentials, even though they turn out to be identical. By using the above new variables, one can transform equation (3.6), with α=0, to the free particle equations, i.e. d2w1/dz2=0 and d2w2/dz2=0.

Unlike the earlier example, one cannot unambiguously integrate these two linear equations in terms of the original variables because of the non-local nature of the independent variable. To overcome this problem, one should express (1+λr2) in terms of either z1 or z2 so that the resultant expression dz1=dz2=dt/(1+λr2) can be integrated. In the following, we describe a procedure to obtain an expression for the new independent variable.

Now integrating equation (3.8) and using the first relation in equation (3.10), we obtain Embedded Image 3.11 where we have fixed the integration constant to be zero (without loss of generality). On the other hand, from expressions (3.8) and (3.9), we obtain Embedded Image, from which one obtains Embedded Image 3.12 where Embedded Image is the integration constant. Equation (3.12) provides us with an identity Embedded Image 3.13 Now squaring and adding equations (3.11) and (3.13), we obtain Embedded Image 3.14 From equation (3.14), one can express (1+λr2) in terms of z1 as Embedded Image 3.15 Substituting equation (3.15) in the last relation given in equation (3.10), we arrive at the following integral relationship between z1 and t, namely Embedded Image 3.16

As the variables are separated now, one can integrate this equation and obtain an expression that connects the new independent variable with the old independent variable through the relation Embedded Image 3.17 where Embedded Image and t0 is the fourth integration constant. From equations (3.11)–(3.13) and (3.15), we obtain Embedded Image 3.18

Substituting expression (3.17) in equation (3.18) and simplifying the resultant expressions, we arrive at the following general solution for equation (3.6), with α=0, in the form: Embedded Image 3.19 where Embedded Image and Embedded Image.

(iii) Example 3: generalized linearizing transformation of type I

In the previous example, we restricted the new independent variable to be a non-local one and a function of only t, x and y. Now we relax the latter condition and also allow the independent variables z1 and z2 to contain derivative terms, namely Embedded Image and Embedded Image. To illustrate this case, let us consider the two coupled second-order equations of the form Embedded Image 3.20 One can easily identify two integrals for equation (3.20) of the form Embedded Image 3.21 Rewriting the above integrals as Embedded Image 3.22 we identify the following set of linearizing transformations for equation (3.20), i.e. Embedded Image 3.23 One may note that the independent variables are not only non-local, but also involve derivative terms. It is easy to check that equation (3.23) transforms equation (3.20) to the linearized form (2.3).

Again, as in the previous example, one cannot directly obtain the solution for equation (3.20) by direct integration of the linear ODEs because of the non-local nature of the independent variables. This can be overcome by expressing the term Embedded Image in terms of either z1 or z2 so that the resultant equation can be integrated to obtain an explicit form for the new independent variable in terms of the old variables, as we discuss below.

From equation (3.21), we have Embedded Image 3.24 Since dw1/dz1=I1 (from equation (3.22)), we have w1=I1z1, so that Embedded Image 3.25 where, without loss of generality, we have fixed the integration constant to be zero. On the other hand, from equation (3.22), we have dw1/dw2=I1/I2, which, in turn, gives Embedded Image 3.26 In terms of old variables, equation (3.26) can be rewritten as Embedded Image 3.27 From the identites (3.25) and (3.27), we can express y/x in terms of z1 in the form Embedded Image 3.28 Now substituting equation (3.28) into equation (3.24), we can express Embedded Image in terms of z1 and, plugging the latter relation into the third relation in equation (3.23), we arrive at Embedded Image 3.29 Integrating the above equation, we obtain Embedded Image 3.30 where t0 is the fourth integration constant. Substituting the expression z1 = (t−1/x)/I1 (equation (3.25)) into equations (3.28) and (3.30), we obtain the general solution of equation (3.20) in the implicit form Embedded Image 3.31

(b) Class B linearizing transformations (z1z2)

In the class A category, in all the three examples, we considered that the new independent variables z1 and z2 are identical. However, this need not always be the case in the theory of linearizing transformations, as discussed in §2. We now present specific examples to illustrate more general transformations.

(i) Example 4: point transformation of type II

Let us consider a quasi-periodic oscillator governed by a set of two coupled SNODEs of the form Embedded Image 3.32

To explore the linearizing transformation for equation (3.32), we consider the two associated integrals Embedded Image 3.33 Rewriting equation (3.33) in the form Embedded Image 3.34 and Embedded Image 3.35 one can identify the new dependent and independent variables as Embedded Image 3.36 One may note that now the independent variables z1 and z2 are not the same.

The new variables transform equation (3.32) to the free particle equations Embedded Image and Embedded Image. From the general solutions w1=I1z1+I3 and w2=I2z2+I4, where Ii, i=1,2,3,4, are the integration constants, and using the expressions for wi and zi, i=1,2, given in equation (3.36), we arrive at the general solution for equation (3.32) in the form Embedded Image 3.37 where Embedded Image and Embedded Image.

(ii) Example 5: mixed point-generalized Sundman transformation

Let us consider the two-dimensional force-free coupled Duffing–van der Pol (DVP) oscillator equation of the form Embedded Image 3.38

One may note that the point transformations X=k1x+k2y and Y =k1xk2y help one to rewrite equation (3.38) in a separable form Embedded Image 3.39a and Embedded Image 3.39b The solution to equation (3.39a) can only be obtained in implicit form (Chandrasekar et al. 2005). Consequently, equation (3.39b) cannot be solved explicitly in this way. Further, the linearization of the scalar DVP oscillator (3.39a) itself has not yet been reported. In the following, we use our procedure to find the linearizing transformation and general solution to equation (3.38) straightforwardly.

The first two integrals for equation (3.38) can easily be identified using the procedure given in Chandrasekar et al. (2009) in the form Embedded Image 3.40 The above integrals can be rewritten as Embedded Image 3.41 from which we can obtain the following linearizing transformations: Embedded Image 3.42 One may note that, in the present problem, one of the new independent variables, i.e. z2, is in a non-local form. In terms of the above new variables, equation (3.38) assumes the forms Embedded Image and Embedded Image.

Now we seek the general solution of equation (3.38) from the linearized equations. Integrating Embedded Image, we obtain Embedded Image 3.43 where I3 is the integration constant. Rewriting equation (3.43) in terms of the old variables, we obtain Embedded Image 3.44 However, the second linear equation, Embedded Image, cannot be integrated straightforwardly (in terms of the original variables) because of the non-local nature of the second independent variable. To obtain an explicit form for z2, we rewrite I2 in the integral form (equation (3.41)) to obtain Embedded Image 3.45 Equation (3.45) provides Embedded Image 3.46 Now substituting relation (3.46) into the non-local variable z2 (equation (3.42)), one obtains Embedded Image 3.47 Solving the above equation, we obtain Embedded Image 3.48 where Embedded Image and t0 is the fourth integration constant. From expression (3.48) and equations (3.44) and (3.46), one can deduce the general solution for equation (3.38) in implicit form. The resultant expression coincides exactly with eqn (6.18) given in Chandrasekar et al. (2009).

In the present example, we considered one of the independent variables to be in a non-local form. As we have two independent variables, one can also have the possibility of having both the independent variables to be of non-local form. Indeed, this is the case in our next example.

(iii) Example 6: generalized Sundman transformation of type II

To illustrate the GST II, we consider the equation of the form Embedded Image 3.49 The first two integrals for equation (3.49) can be evaluated as Embedded Image 3.50 Rewriting these two integrals as Embedded Image 3.51 we identify the linearizing transformations in a more general form Embedded Image 3.52 The GST II equation (3.52) takes equation (3.38) to the free particle equations, Embedded Image and Embedded Image. To obtain the solution in terms of the original variables, we have to replace both Embedded Image and Embedded Image by the variables z1 and t, and z2 and t, respectively, and integrate the resultant equations.

To do so, first we rewrite the first integrals I1 and I2 given by equation (3.51) in integral forms, which in turn lead us to w1=I1z1 and w2=I2z2. As w1 and w2 do not contain non-local variables, we can replace them by the old variables (equation (3.52)), i.e. Embedded Image 3.53 where we have fixed the integration constants to be zero (without loss of generality).

We observe that, to integrate the last two expressions in equation (3.52), one should further replace z1 and z2 in terms of t. So, we substitute the above expressions for x and y in terms of z1 and z2, respectively, in the last two relations in equation (3.52), and obtain Embedded Image 3.54 Now integrating both the equations, we obtain Embedded Image 3.55 where I3 and I4 are the third and forth integration constants, respectively. Plugging equation (3.55) into equation (3.53), we arrive at the following general solution for equation (3.49): Embedded Image 3.56 where Embedded Image and Embedded Image.

In the previous two examples, we focussed our attention on the case in which the new independent variable(s) is (are) non-local and does (do) not admit velocity-dependent terms. Now we relax this condition and allow either one or both the independent variables to admit velocity-dependent terms but in non-local form.

(iv) Example 7: mixed point-generalized linearizing transformation

To demonstrate this, we consider a variant of the two-dimensional Mathews and Lakshmanan equation (3.6) of the form Embedded Image 3.57 Equation (3.57) admits the following two integrals of motion: Embedded Image3.58 Rewriting equation (3.58) in the form Embedded Image 3.59 and Embedded Image 3.60 one can easily identify the linearizing transformations for equation (3.57) as Embedded Image 3.61 In terms of the above new variables, equation (3.57) gets transformed to the free particle equations (2.3). One may note that one of the new independent variables is not only in non-local form, but also contains derivative terms that, in turn, complicate the situation to obtain the general solution.

As both w1 and z1 are of point transformation types, one can integrate the first free particle equation, namely Embedded Image and obtain w1=I1z1+I3, where I3 is an integration constant. Replacing the latter in terms of the old variables, one obtains the relation (xy)=I1t+I3. On the other hand, from the solution of the second linear equation Embedded Image, we can write Embedded Image (again we assume the integration constant to be zero without loss of generality).

To evaluate z2, let us first substitute equation (3.58) into equation (3.61) and rewrite the latter in the form Embedded Image3.62 Now substituting the form of Embedded Image (equation (3.58)), i.e. Embedded Image 3.63 into equation (3.62) and using the relation (1+2λ(x+y))=eI2z2, we obtain Embedded Image 3.64 Integrating equation (3.64), we obtain Embedded Image 3.65 where t0 is an integration constant. Substituting expression (3.65) into the relation 2λ(x+y)=eI2z2−1, we obtain Embedded Image3.66 From equation (3.66) and the relation (xy)=I1t+I3, we obtain the general solution for equation (3.57) in the form Embedded Image3.67 In this example, we considered the case in which only one of the independent variables is in non-local form. Now we consider the case in which both the independent variables are in non-local forms.

(v) Example 8: mixed generalized Sundman-generalized linearizing transformation

To illustrate this type of linearizing transformation, let us again consider equation (3.6), but now with α≠0. Equation (3.6) admits the following two integrals of motion: Embedded Image3.68 Rewriting these two integrals in the form Embedded Image 3.69 and Embedded Image 3.70 and identifying the linearizing transformations, we obtain Embedded Image 3.71 One can check that equation (3.71) transforms equation (3.6) to the form of equation (2.3).

Rewriting the first integrals I1 and I2 in the integral form and identifying them in terms of the new variables, we have w1=I1z1 and w2=I2z2 that, in turn, also give us a relationship between x and y with z1 and z2, respectively (after fixing the integration constants to be zero without loss of generality), i.e. Embedded Image 3.72 Expressing Embedded Image and Embedded Image in terms of I1,I2,x and y (by using equation (3.68)) and substituting them in the expression for dz2, we obtain Embedded Image 3.73 Now, from the expression for 1+λr2 (equation (3.72)), we obtain Embedded Image3.74 Integrating the above equation, we obtain Embedded Image3.75 where I3 is an integration constant that is nothing but the third integral of motion. In order to find the fourth integral, using dz1=dt/y2 and equation (3.74), we eliminate t to obtain Embedded Image3.76 From equation (3.72), we obtain Embedded Image, which on substitution into equation (3.76) leads to Embedded Image3.77 Now integrating equation (3.77), we obtain Embedded Image 3.78 where I4 is the fourth integration constant. Now making use of these four integrals of motion, namely equations (3.68), (3.75) and (3.78), the general solution to equation (3.6) can be straightforwardly constructed. The resultant solution also agrees with eqn (5.40) of Chandrasekar et al. (2009), obtained through the modified PS approach, after a redefinition of integration constants.

(vi) Example 9: generalized linearizing transformation of type II

To understand the generalized linearizing transformation, let us start with the following system of coupled second-order ODEs: Embedded Image 3.79 The associated first integrals are Embedded Image 3.80 Rewriting equation (3.80) in the form Embedded Image 3.81 and identifying the new variables, we obtain the linearizing transformation Embedded Image 3.82

From the first integrals, we have (after assuming the integration constants to be zero without loss of generality) Embedded Image 3.83 Using equation (3.82) in equation (3.83), we obtain Embedded Image 3.84 Substituting the expressions of x and y into equation (3.80) and solving the resultant equation for Embedded Image and Embedded Image, we obtain Embedded Image 3.85 Substituting equations (3.84) and (3.85) into the expressions (3.82) for dz1 and dz2 and integrating the resultant equation, we obtain Embedded Image 3.86 where I3 and I4 are the third and fourth integration constants, respectively. From equations (3.84) and (3.86), we can obtain the general solution for equation (3.79) straightforwardly.

4. Conclusions

In this paper, we have studied the linearization of two coupled SNODEs. In particular, we have introduced a new method of deriving linearizing transformations from the first integrals for the given equation. The procedure is simple and straightforward. From our analysis, we have demonstrated that one can have two wider classes of linearizing transformations, namely class A and class B, depending on the nature of the independent variables. In class A category, the independent variables are the same, and we identified three types of linearizing transformations in which two of them are new to the literature. On the other hand, in the class B category (the independent variables are different), we found six new types of linearizing transformations. We have explicitly demonstrated the method of deducing the linearizing transformations and the general solution for all of these cases with specific examples. However, in this paper, we have restricted our attention to two aspects: (i) dependent variables are functions of only (t,x,y) and (ii) independent variables are not of the local form Embedded Image, where i=1,2. Linearization under these two types requires separate treatment and will be studied subsequently. The method proposed here can naturally be extended to any number of coupled second-order ODEs and indeed one can derive a very wide class of linearizing transformations in these cases.

Acknowledgements

The work of M.S. forms part of a research project sponsored by the National Board for Higher Mathematics, Government of India. The work of M.L. forms part of a Department of Science and Technology, Government of India, sponsored research project and was supported by a DST Ramanna Fellowship.

Footnotes

    • Received January 24, 2009.
    • Accepted April 7, 2009.

References

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