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A systematic method of finding linearizing transformations for nonlinear ordinary differential equations: I. Scalar case
TL;DR: In this article, the maximal number of linearizing transformations for nonlinear ODEs of any order including coupled ones from a knowledge of fewer number of integrals of motion is derived.
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Abstract: In this set of papers we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and efficient and helps to unearth several new types of linearizing transformations besides the known ones in the literature. To make our studies systematic we divide our analysis into two parts. In the first part we confine our investigations to the scalar ODEs and in the second part we focuss our attention on a system of two coupled second order ODEs. In the case of scalar ODEs, we consider second and third order nonlinear ODEs in detail and discuss the method of deriving maximal number of linearizing transformations irrespective of whether it is local or nonlocal type and illustrate the underlying theory with suitable examples. As a by-product of this investigation we unearth a new type of linearizing transformation in third order nonlinear ODEs. Finally the study is extended to the case of general scalar ODEs. We then move on to the study of two coupled second order nonlinear ODEs in the next part and show that the algorithm brings out a wide variety of linearization transformations. The extraction of maximal number of linearizing transformations in every case is illustrated with suitable examples.
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On the symmetry solutions of two-dimensional systems not solvable by standard symmetry analysis
TL;DR: In this paper, a class of systems of second-order ODEs is identified in which a system requires fewer Lie point symmetries than required to solve it, and the procedure distinguishes among those which are linearizable, complex-linearizable and solvable systems.
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A note on deriving linearizing transformations for a class of second order nonlinear ordinary differential equations
TL;DR: In this article, a general form of a nonlinear ODE that admits Bernoulli equation as its first integral is constructed, and conditions for this integral to yield three different linearizing transformations, namely point, Sundman and generalized linearizing transformation.
1
Periodic Solutions of a Class of Second-Order Differential Equation
TL;DR: In this paper, the authors studied the periodic solutions of second-order differential equations of the form where the functions,, and are periodic of period in the variable t, where t is a constant.
Linearization from Complex Lie Point Transformations
TL;DR: A geometrical construction of the procedure adopted that provides an analogue in of the linearizability criteria in .
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Hamiltonian structures for the n‐dimensional Lotka–Volterra equations
TL;DR: Under what algebraic conditions n−dimensional Lotka-Volterra equations are Hamiltonian for a suitable Poisson structure is investigated in this article, where a poisson structure for a Poisson Poisson is proposed.
155
On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations
TL;DR: In this article, a method for finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle-Singer (PS) method is briefly discussed, and the general solution associated with several dynamical systems discussed in the current literature by employing their modifications and extensions of the PS method.
A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators
TL;DR: In this paper, a super-integrable two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms, where all the bounded motions are quasiperiodic oscillations and the unbounded motions are represented by hyperbolic functions.
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First integrals, integrating factors and λ-symmetries of second-order differential equations
C. Muriel,J. L. Romero +1 more
TL;DR: For a given second-order ordinary differential equation (ODE), several relationships among first integrals, integrating factors and λ-symmetries are studied in this paper, and an algorithm to find two functionally independent first integral is provided.
121
On the complete integrability and linearization of certain second order nonlinear ordinary differential equations
TL;DR: In this article, a method of finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle-Singer (PS) method is briefly discussed, and the authors explore integrating factors, integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing their modifications and extensions of the PS method.
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