A method for finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle–Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the PS method. We also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second-order nonlinear ordinary differential equations to free particle equations. We illustrate the theory with several potentially important examples and show that our procedure is widely applicable.

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On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations

A Lienard type nonlinear oscillator of the form x+kxx+(k2/9)x3+lambda1x=0, which may also be considered as a generalized Emden-type equation, is shown to possess unusual nonlinear dynamical properties. It is shown to admit explicit nonisolated periodic orbits of conservative Hamiltonian type for lambda1>0. These periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator. This is completely contrary to the standard characteristic property of nonlinear oscillators. Interestingly, the system though appears deceptively a dissipative type for lambda1< or =0 does admit a conserved Hamiltonian description, where the characteristic decay time is also independent of the amplitude. The results also show that the criterion for conservative Hamiltonian system in terms of divergence of flow function needs to be generalized.

Unusual Liénard-type nonlinear oscillator

The existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riccati equation. The Lagrangians are non-natural and the forces are not derivable from a potential. The constant value E of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems. In the second part the existence of two-dimensional versions endowed with superintegrability is proved. The explicit expressions of the additional integrals are obtained in both cases. Finally it is proved that the orbits of the second system, that represents a nonlinear oscillator, can be considered as nonlinear Lissajous figures

Lagrangian formalism for nonlinear second-order Riccati systems: One-dimensional integrability and two-dimensional superintegrability

For a given second-order ordinary differential equation (ODE), several relationships among first integrals, integrating factors and λ-symmetries are studied The knowledge of a λ-symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations If two nonequivalent λ-symmetries of the equation are known, then an algorithm to find two functionally independent first integrals is provided These methods include and complete other methods to find integrating factors or first integrals that are based on variational derivatives or in the Prelle–Singer method These results are applied to several ODEs that appear in the study of relevant equations of mathematical physics

First integrals, integrating factors and λ-symmetries of second-order differential equations

This is the second in a series of two tutorial articles devoted to triangulation-decomposition algorithms. The value of these notes resides in the uniform presentation of triangulation-decomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. The present article deals with differential systems. It uses results presented in the first article on polynomial systems but can be read independently.

Notes on triangular sets and triangulation-decomposition algorithms II: Differential systems

We propose a method for solving second-order ordinary differential equations (ODEs) which is based on the ideas behind the Prelle-Singer (PS) procedure for first-order ODEs. While the PS procedure treats differential equations (DEs) of the form y' = P(x,y)/Q(x,y), with P and Q polynomials whose coefficients lie in the field of complex numbers , our method is applicable to DEs of the form y'' = P(x,y,y')/Q(x,y,y'). The key to our approach is to focus not on the final solution but on the first-order invariants of the equation. Our method is an attempt to address algorithmically the solution of second-order ODEs with solutions in terms of elementary functions.

https://iopscience.iop.org/article/10.1088/0305-4470/34/14/308/pdf

Solving second-order ordinary differential equations by extending the Prelle-Singer method

http://cds.cern.ch/record/307131/files/9607037.pdf

Computer algebra solving of first order ODEs using symmetry methods

Here we demonstrate a theorem concerning the general structure of the integrating factor for first-order ordinary differential equations whose solutions contain Liouvillian functions. This result assures the generality of a method presented in a forthcoming paper extending the usual Prelle-Singer approach.

https://iopscience.iop.org/article/10.1088/0305-4470/35/4/312/pdf

Analysing the structure of the integrating factors for first-order ordinary differential equations with Liouvillian functions in the solution

The usual Prelle–Singer (PS) approach misses many first-order ordinary differential equations presenting Liouvillian functions in the solution (LFOODEs). We point out why and propose a method extending the PS method to solve a class of these previously unsolved LFOODEs. Although our method does not cover all the LFOODEs, it maintains the semi-decision nature of the usual PS method.

https://iopscience.iop.org/article/10.1088/0305-4470/35/17/306/pdf

L.A.C.P. da Mota

Papers

Solving second-order ordinary differential equations by extending the Prelle-Singer method

Computer algebra solving of first order ODEs using symmetry methods

Analysing the structure of the integrating factors for first-order ordinary differential equations with Liouvillian functions in the solution

A method to tackle first-order ordinary differential equations with Liouvillian functions in the solution

An extension of the Prelle–Singer method and a Maple implementation☆