We demonstrate a simplification of some recent works on the classification of the Lie symmetries for a quadratic equation of Liénard type. We observe that the problem could have been resolved more simply.
In Refs. 1 and 2 the classification of the Lie (point) symmetries for the quadratic equation of the form
was performed, where overdot denotes total differentiation with respect to time, “t.”
We observe that under the coordinate (not point) transformation,
Equation (1) takes the simpler form
where
The classification of the Lie symmetries of the latter equation has been known for some time (approximately 140 yr). The various possibilities are as follows.
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is an arbitrary function. Equation (3) admits the autonomous symmetry, ∂t, and the equation can be simply reduced to a first integral which in general cannot be evaluated to obtain the solutions in closed form.
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In the two cases
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and
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the admitted Lie symmetries of (3) constitute the algebra A2 in the Mubarakzyanov classification scheme.3–6
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When
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or
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, β ≠ 0,
Equation (3) is invariant under the three-dimensional algebra, A3,8, which is more commonly known as .
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Finally for the cases
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,
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,
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, and
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(note that in (c) and (d) a multiplicative constant—or arbitrary function of time which is beyond the considerations of Refs. 1 and 2—in the y term is superfluous), the algebra of the Lie symmetries is and, as a second-order linear equation, (3) is maximally symmetric [Ref. 7, p. 405].
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A.P. acknowledges Professor P. G. L. Leach, Sivie Govinder, as also DUT for the hospitality provided and the UKZN, South Africa, for financial support. The research of A.P. was supported by FONDECYT postdoctoral Grant No. 3160121.