We investigate a class of Lagrangians that admit a type of perturbed harmonic oscillator which occupies a special place in the literature surrounding perturbation theory. We establish explicit and generalized geometric conditions for the symmetry determining equations. The explicit scheme provided can be followed and specialized for any concrete perturbed differential equation possessing the Lagrangian. A systematic solution of the conditions generate nontrivial approximate symmetries and transformations. Detailed cases are discussed to illustrate the relevance of the conditions, namely (a) G 1 as a quadratic polynomial, (b) the Klein-Gordon equation of a particle in the context of Generalized Uncertainty Principle and (c) an orbital equation from an embedded Reissner-Nordström black hole.

In this paper, all the Lie point symmetries of difference equations of the form u n+4 = u n A n + B n u n u n+2 , where, (A n) n 0 and (B n) n 0 are sequences of real numbers, are obtained. We perform reduction of order using the invariant of the group of transformations. Furthermore, we obtain their solutions. In particular, our work generalizes some results in the literature.

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A full Lie point symmetry analysis of rational difference equations is performed. Non-trivial symmetries are derived and exact solutions using these symmetries are obtained.

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