A unification in the theory of linearization of second-order nonlinear ordinary differential equations

In this letter, we introduce a new generalized linearizing transformation (GLT) for second-order nonlinear ordinary differential equations (SNODEs). The well-known invertible point (IPT) and non-point transformations (NPT) can be derived as sub-cases of the GLT. A wider class of nonlinear ODEs that cannot be linearized through NPT and IPT can be linearized by this GLT. We also illustrate how to construct GLTs and to identify the form of the linearizable equations and propose a procedure to derive the general solution from this GLT for the SNODEs. We demonstrate the theory with two examples which are of contemporary interest. PACS numbers: 02.30.Hq, 05.45.−a Linearizing nonlinear ordinary differential equations (NODEs) is still an open problem in the theory of differential equations [1–3]. If one raises the question whether a given arbitrary nonlinear ODE is linearizable or not, no definitive answer can be given in general. Three main points which need attention for further understanding of this problem are: (i) there is still no comprehensive literature available on the types of transformations that can linearize the ODEs, (ii) the general form of linearizable equation also differs from transformation to transformation and (iii) higher-order ODEs possess a greater variety of linearizing transformations than the lower-order ODEs. Due to these reasons no general treatment on linearizing transformations or linearizable equations has been formulated so far. In this letter, we make an attempt to unify the linearizing transformations known for the case of second-order nonlinear ODEs (SNODEs) and extend their scope. As far as the SNODEs are concerned it has been shown that, in general, one can linearize them through two different kinds of transformations. One is the well-known invertible point transformation (IPT) and the other is the non-point transformation (NPT). As far as the IPT is concerned it has been shown [3–8] that the most general SNODE that can be linearized through such a transformation, X = F(t, x), T = G(t, x), (1) 0305-4470/06/030069+08$30.00 © 2006 IOP Publishing Ltd Printed in the UK L69 L70 Letter to the Editor is of the form ẍ = D(t, x)ẋ + C(t, x)ẋ + B(t, x)ẋ + A(t, x), (2) where over dot denotes differentiation with respect to t and the functions A,B,C and D should satisfy the following two equations: 3Dtt + 3BDt − 3ADx + 3DBt + Bxx − 6DAx + CBx − 2CCt − 2Ctx = 0, Ctt + 6ADt − 3ACx + 3DAt − 2Btx − 3CAx + 3Axx + 2BBx − BCt = 0. (3) The transformation (1) converts equation (2) into the linear ‘free particle’ equation, dX dT 2 = 0. (4) On the other hand, it has also been shown that one can consider NPTs of the form X = F̂ (t, x), dT = Ĝ(t, x) dt, (5) and linearize the given SNODE. The most general SNODE that can be linearized through the transformation (5) possess the form [9] ẍ + A2(t, x)ẋ 2 + A1(t, x)ẋ + A0(t, x) = 0. (6) The set of relations between the functions Ai’s, i = 0, 1, 2, and the transformation (5) is given by A2 = (ĜF̂ xx − F̂ xĜx)/K, A1 = (2ĜF̂ xt − F̂ xĜt − F̂ t Ĝx)/K, (7) A0 = (ĜF̂ tt − F̂ t Ĝt )/K with K = F̂ xĜ = 0. The NPT also transforms equation (6) to the free particle equation (4). The functions Ai’s, i = 0, 1, 2, should satisfy the following relations [9]: (i) S1(t, x) = A1x − 2A2t = 0, (8) S2(t, x) = 2A0xx − 2A1tx + 2A0A2x − A1xA1 + 2A0xA2 + 2A2t t = 0. (9) (ii) If S1(t, x) = 0 and S2(t, x) = 0, then S 2 + 2S1t S2 − 2S 1A1t + 4S 1A0x + 4S 1A0A2 − 2S1S2t − S 1A1 = 0, (10) S1xS2 + S 2 1A1x − 2S 1A2t − S1S2x = 0. (11) The NPT is also called a generalized Sundman transformation, see for example [10, 11]. Even though both the IPT and NPT transform the second-order nonlinear ODE to the free particle equation (4), the NPT has some disadvantages over the former. For example, in the case of IPT one can unambiguously invert the free particle solution and deduce the solution of the associated nonlinear equation, whereas in the case of NPT it is not so straightforward due to the non-local nature of the independent variable. In this work, we unearth a more general transformation, X = F(t, x), dT = G(t, x, ẋ) dt, (12) and show that this transformation can be utilized to linearize a wider class of SNODEs and, in particular, certain equations which cannot be linearized by the NPT and IPT. We designate this transformation as the generalized linearizing transformation (GLT). If the function G in (12) is independent of the variable ẋ then it becomes an NPT (vide equation (5)). On the other hand, if G is a perfect differentiable function then it becomes an IPT, that is G(t, x, ẋ) = d dt Ĝ(t, x), Letter to the Editor L71 then dT = dĜ dt dt ⇒ T = Ĝ(t, x). We stress here that (12) is a unified transformation as it includes IPT and NPT as special cases. We demonstrate our above assertion with the case where G is a polynomial function in ẋ and in particular where it is linear in ẋ with coefficients which are arbitrary functions of t and x. Indeed, even such a simple case leads to interesting results as we see below. To be specific, we focus here on the case X = F(t, x), dT = (G1(t, x)ẋ + G2(t, x)) dt. (13) We note that in equation (13) even if we considerX = F(t, x, ẋ) and dT = (G1ẋ+G2) dt , after substitution into (16), we deduce that Fẋ = 0 and so the form (13) is taken. Generalizations involving higher degree polynomials in ẋ for G(t, x, ẋ) will be dealt with elsewhere. Substituting the transformation (13) into the free particle equation (4), the most general SODE that can be linearized through the GLT (13) can be shown to be of the form ẍ + A3(t, x)ẋ 3 + A2(t, x)ẋ 2 + A1(t, x)ẋ + A0(t, x) = 0 (14) and the functions Ai’s i = 0, 1, 2, 3, are connected to the transformation functions F and G through the relations A3 = (G1Fxx − FxG1x)/M, A2 = (G2Fxx + 2G1Fxt − FxG2x − FtG1x − FxG1t )/M, A1 = (2G2Fxt + G1Ftt − FxG2t − FtG2x − FtG1t )/M, A0 = (G2Ftt − FtG2t )/M (15) with M = FxG2 − FtG1 = 0. For the given equation one has explicit forms for the functions Ai’s. Now solving equation (15) with the known Ai’s, one can get the linearizing transformation functions F and G. Once F and G are known then using (13) we can transform (14) to the free particle equation (4) and solving the latter one can get the first integral. However, it is difficult to integrate it further unambiguously to obtain the general solution due to the non-local nature of the transformation (13). We are able to overcome this problem also here and devise a general procedure to construct the general solution. In the following, we briefly describe the procedure. Integrating the free particle equation (4) once, we get dX dT = I1 = C(t, x, ẋ), (16) where I1 is the first integral. Now rewriting (16) for ẋ, we get ẋ = f (t, x, I1), (17) where f is a function of the indicated variables. Due to non-local nature of the independent variable we need to consider only a particular solution for the free particle equation (4), that is X(t, x) = I1T (18) from which we get x = g(t, T , I1), (19) where g is a function of t, T and I1. Making use of relations (17) and (19), equation (13) can be rewritten in the form dT = h(t, T , I1) dt, (20) L72 Letter to the Editor where again h is a function of t, T and I1. We find that in the case of linearizable equations one can separate the variables T and t in equation (20) and integrate the resultant equation which in turn leads to the general solution. In the above, we have demonstrated how to deduce linearizing transformation and the general solution for the given equation. On the other hand, one can construct both linearizing transformation and specific linearizable equations. To illustrate this let us analyse a particular but important case of equation (14), namely, A3 = 0 and A2 = 0 in equation (15). However, the other choices, for example A3 = A1 = 0 and A2 and A0 = 0, also lead to many new linearizable equations. These will be dealt with separately. Solving the first and second equation in (15) with this restriction, we obtain G1 = a(t)Fx, G2 = a(t)Ft − (atx + b(t))Fx, (21) where a and b are arbitrary functions of t. By using equation (21) in the last two equations in (15), we get A1 = Sx + at (atx + b) S + (att x + bt ) (atx + b) , (22) A0 = St + at (atx + b) S + (att x + bt ) (atx + b) S, (23) where S(t, x) = Ft Fx . (24) Solving equation (22), we get S = ( c(t)− btx − 2attx + ∫ A1(atx + b) dx )