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Original Articles

A Note on the Integrability of a Class of Nonlinear Ordinary Differential Equations

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Pages 159-164 | Published online: 21 Jan 2013

References

  • Andriopoulos , K and Leach , PGL . 2006 . An interpretation of the presence of both positive and negative nongeneric resonances in the singularity analysis . Phys. Lett. A , 359 : 199 – 203 .
  • Camiz , P , Geradi , A , Marchioro , C , Presutti , E and Scacciatelli , E . 1971 . Exact solution of a time-dependent quantal harmonic oscillator with a singular perturbation . J. Math. Phys. , 12 : 2040 – 2043 .
  • Conte , R . 1994 . “ Singularities of differential equations and integrability ” . In Introduction to Methods of Complex Analysis and Geometry for Classical Mechanics and Nonlinear Waves Edited by: Behest , D , Froeschlé , C and Éditions , Frontières . 49 – 143 . Gif-sur-Yvette
  • Ermakov , V . 1880 . Second-order differential equations. Conditions of complete integrability . University Izvestia Kiev Series III , 9 : 1 – 25 . trans Harin AO
  • Euler , N and Leach , PGL . 2003 . First integrals and reduction of a class of nonlinear higher order ordinary differential equations . J. Math. Anal. Appl. , 287 : 337 – 347 .
  • Leach , PGL . 1977 . On a direct method for the determination of an exact invariant for the time-dependent harmonic oscillator . J. Aust. Math. Soc., Series B , 20 : 97 – 105 .
  • Leach , PGL . 1977 . Invariants and wave-functions for some time-dependent harmonic oscillator-type Hamiltonians . J. Math. Phys. , 18 : 1902 – 1907 .
  • Leach , PGL . 1990 . Berry’s phase and wave functions for time-dependent Hamiltonian systems . J.Phys.A: Math. Gen. , 23 : 2695 – 2699 .
  • Leach , PGL and Govinder , KS . Hidden symmetries and integration of the generalized Emden-Fowler equation of index two . Proc. 14th IMACS World Congress on Comput. App. Math . Edited by: Ames , WF . pp. 300 – 303 . Atlanta : Georgia Institute of Technology .
  • Lewis , HR JR . 1967 . Classical and quantum systems with time-dependent harmonic oscillator-type Hamiltonians . Phys. Rev. Lett. , 18 : 510 – 512 .
  • Lewis , HR JR . 1968 . Motion of atime-dependent harmonic oscillator and of a charged particle in a time-dependent, axially symmetric, electromagnetic field . Phys. Rev. , 172 : 1313 – 1315 .
  • Lewis , HR JR . 1968 . Class of exact invariants for classical and quantum time-dependent harmonic oscillators . J. Math. Phys. , 9 : 1976 – 1986 .
  • Lewis , H , Ralph , JR and Riesenfeld , WB . 1969 . An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in atime-dependent electromagnetic field . J. Math. Phys. , 10 : 1458 – 1473 .
  • Moyo , S and Leach , PGL . Reduction properties of ordinary differential equations of maximal symmetry . Proc. Global Integrability of Field Theories (GIFT 2006) . pp. 253 – 266 . Daresbury, UK : Cockcroft Institute . ISBN: 3–86644–035–9
  • Pinney , E . 1980 . The nonlinear differential equation y″(x) + p(x)y + cy −3 = 0 . Proc. Amer. Math. Soc. , 1 : 681
  • Ramani , A , Grammaticos , B and Bountis , T . 1989 . The Painlevé property and singularity analysis of integrable and nonintegrable systems . Phys. Rep. , 180 : 159 – 245 .
  • Tabor , M . 1989 . Chaos and Integrability in Nonlinear Dynamics , New York : John Wiley .
  • Whittaker , ET . 1944 . A Treatise on the Analytical Dynamics of Particles and Rigid Bodies , New York : Dover .

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