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A simple and unified approach to identify integrable nonlinear oscillators and systems. (English) Zbl 1111.34003
Summary: We consider a generalized second-order nonlinear ordinary differential equation (ODE) of the form $\stackrel{¨}{x}+\left({k}_{1}{x}^{q}+{k}_{2}\right)\stackrel{˙}{x}+{k}_{3}{x}^{2q+1}+{k}_{4}{x}^{q+1}+{\lambda }_{1}x=0$, where ${k}_{i}$, $i=1,2,3,4$, ${\lambda }_{1}$, and $q$ are arbitrary parameters, which includes several physically important nonlinear oscillators such as the simple harmonic oscillator, anharmonic oscillator, force-free Helmholtz oscillator, force-free Duffing and Duffing-van der Pol oscillators, modified Emden-type equation and its hierarchy, generalized Duffing-van der Pol oscillator equation hierarchy, and so on, and investigate the integrability properties of this rather general equation. We identify several new integrable cases for arbitrary value of the exponent $q$, $q\in ℝ$. The cases $q=1$ and $q=2$ are analyzed in detail and the results are generalized to arbitrary $q$. Our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature. To explore the above underlying results, we use the recently introduced generalized extended Prelle-Singer procedure applicable to second-order ODEs. As an added advantage of the method, we not only identify integrable regimes but also construct integrating factors, integrals of motion, and general solutions for the integrable cases, wherever possible, and bring out the mathematical structures associated with each of the integrable cases.
##### MSC:
 34A05 Methods of solution of ODE 34C14 Symmetries, invariants (ODE) 34C15 Nonlinear oscillations, coupled oscillators (ODE) 70K99 Nonlinear dynamics (general mechanics)