We respond to the comment on ‘Classification of Lie point symmetries for quadratic Liénard type equation + f(x)2 + g(x) = 0’ [J. Math. Phys. 61, 044101 (2013)] by Iacono regarding linearizability and isochronicity. We assert here that the condition for linearization of the equation + f(x)2 + g(x) = 0 given by us in our paper is correct with the condition g1=ω02>0. We present the explicit form of local and nonlocal transformations that transform the quadratic Liénard equation ẍ+F+11xẋ2+x(1x)(1+Dx)=0 into the harmonic oscillator equation for the four cases mentioned in the comment and confirm the statements given in our paper are all valid.

In the paper by Tiwari et al.,1 four of the present authors have classified all the Lie point symmetries admitted by the second order ordinary nonlinear differential equation, namely, the quadratic Liénard equation (LE),

(1)

where f(x) and g(x) are arbitrary functions of x. As an incidental result, we have also pointed out that the subclass of systems possessing maximal Lie point symmetries, which are also mappable by local coordinate transformation to the linear harmonic oscillator,

(2)

(where X is now a function of x only) so that system (1) exhibits isochronous property globally will satisfy condition (38) of our paper,1 

(3)

Here, the arbitrary constant g1=ω02, and obviously for periodic solutions to exist, one requires ω02>0 in (2). There is no restriction on the sign of the arbitrary constant g2. The above results have further been used in Ref. 2.

We also pointed out that the condition for linearization of (1) under point transformation involving coordinate variables only is also exactly the same as that of (3), as given in condition (42) of our paper.1 Although it is implied that the arbitrary constant g1 obtained by solving the consistency condition for linearizability, namely, gxx + fgx + gfx = 0, should be positive definite (>0) for isochronicity (g1=ω02) in (3), we have failed to mention this explicitly in our paper and this should be amended. Obviously, when ω02=0 or <0 in Eq. (2), no periodic solution exists and so no isochronicity occurs in these cases and, consequently, the cases g1 = 0 or <0 should not be included for isochronicity.

Due to the above reasons, we completely agree with the comment of Iacono3 that for sufficient condition of isochronicity as noted in his paper, his Eq. (1) should be K > 0, that is, g1 > 0 in our paper. However, we disagree with his claim, point No. 2 in his comment, that the equation

(4)

is isochronous for four parametric choices of the constant parameters

(5)

under local point transformation. Obviously, the first two choices do satisfy the isochronicity condition (4) with g1 = 1, and the associated differential equations get linearized to (2) under local point transformations, X=xx1, τ = t for (D, F) = (0, 1) and X=121(1x)21, τ = t for (D,F)=12,2. However, the other two cases become linearizable to harmonic oscillator equation (2) but with τt only under nonlocal transformations as we show below.

The case (D,F)=0,14 corresponds to the differential equation

(6)

and gets linearized (see below) to the harmonic oscillator equation (2) only under the nonlocal transformation

(7)

Similarly, the case (D,F)=(12,12) corresponding to the following equation:

(8)

gets transformed to the linear harmonic oscillator equation (2) under the nonlocal transformation

(9)

Thus, as long as local point transformations are allowed, only the first two cases in the parametric choice (5) are allowed satisfying Eq. (3). Therefore, the results pointed out in our papers1,2 stand vindicated, and there is essentially no inadmissible statement that is made in our papers, contrary to what is claimed in the comment of Iacono, subject to the correction that the constant g1 > 0, which is implied in our paper, should be explicitly stated as mentioned above.

It is also easy to find the general transformation for isochronicity associated with nonpoint transformation. We note that Eq. (1) can be integrated once to get the first integral as

(10)

Defining now the transformations

(11)

and

(12)

one can map (10) into

(13)

corresponding to the integral of (2) in the general case τt. In the special case, τ = t, Eq. (8) leads to condition (3). Otherwise, (7) and (9) constitute the most general transformation to isochronous oscillations, which is a nonlocal transformation. In our papers,1,2 we considered only point transformations and all our statements are concerned with such transformations only.

Now, considering comment 5 of Iacono,3 let us consider the LE equation with position dependent linear damping,

(14)

Equation (14) can be linearized to (2) through the point transformation (now involving both x and t variables),

(15)

where z=λsinλt+kxcosλtλkx. One can note that Eq. (14) can also be linearized to (2) through the nonlocal transformation of the form

(16)

leading to the elegant solution

(17)

where A and δ are arbitrary constants, demonstrating the isochronous property as pointed by three of us in Ref. 4.

We feel that the remaining part of the comment is obvious and there is nothing new to be stated.

The work of V.K.C. was supported by the SERB-DST-MATRICS (Grant No. MTR/2018/000676). M.L. was supported by DST-SERB through a Distinguished Fellowship (Grant No. SB/DF/04/2017). The work of M.S. forms part of a research project sponsored by the National Board for Higher Mathematics (NBHM), Government of India, under Grant No. 02011/20/2018NBHM(R.P)/R&D II/15064.

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