First integrals for the damped Helmholtz oscillator

References

• Ablowitz , M. J. and Clarkson , P. A. 1992 . Solitons, Nonlinear Evolution Equations and Inverse Scattering , Cambridge : Cambridge University Press .
   Google Scholar
• Abramowitz , M. and Stegun , I. A. 1970 . Handbook of Mathematical Functions , New York : Dover .
   Google Scholar
• Almendral , J. A. and Sanjuán , M. A.F. 2003 . Integrability and symmetries for the Helmholtz oscillator with friction . J. Phys. A (Math. Gen.) , 36 : 695 – 710 .
   Google Scholar
• Arnold , V. I. 1978 . Mathematical Methods of Classical Mechanics , New York : Springer .
   Google Scholar
• Bluman , G. W. and Anco , S. C. 2002 . Symmetry and Integration Methods for Differential Equations , New York : Springer .
   Google Scholar
• Chandrasekar , V. K. , Senthilvelan , M. and Lakshmanan , M. 2004 . New aspects of integrability of force-free Duffing-van der Pol oscillator and related nonlinear systems . J. Phys. A (Math. Gen.) , 37 : 4527 – 4534 .
   Google Scholar
• Chandrasekar , V. K. , Senthilvelan , M. and Lakshmanan , M. 2005 . On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations . Proc. R. Soc. Lond. Ser. A , 461 : 2451 – 2476 .
   Web of Science ®Google Scholar
• Chandrasekar , V. K. , Senthilvelan , M. and Lakshmanan , M. 2006 . On the complete integrability and linearization of nonlinear ordinary differential equations–(II) third-order equations . Proc. R. Soc. Lond. Ser. A , 462 : 1831 – 1852 .
   Web of Science ®Google Scholar
• Duarte , L. G.S. , Duarte , S. E.S. , da Mota , A. C.P. and Skea , J. E.F. 2001 . Solving the second-order ordinary differential equations by extending the Prelle–Singer method . J. Phys. A (Math. Gen.) , 34 : 3015 – 3024 .
   Google Scholar
• Feng , Z. 2003 . Travelling wave solutions and proper solutions to the two-dimensional Burgers–Korteweg–de Vries equation . J. Phys. A (Math. Gen.) , 33 : 8817 – 8827 .
   Google Scholar
• Feng , Z. and Meng , Q. G. 2007 . Exact solution for a two-dimensional KdV-Burgers-type equation with nonlinear terms of any order . Discrete Contin. Dyn. Syst. Ser. B , 7 : 285 – 291 .
   Google Scholar
• Hsu , S. B. 2006 . Ordinary Differential Equations with Applications , New Jersey : World Scientific .
   Google Scholar
• Ince , E. L. 1956 . Ordinary Differential Equations , New York : Dover .
   Google Scholar
• Jordan , D. W. and Smith , P. 2007 . Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers , New York : Oxford University Press .
   Google Scholar
• Lie , S. 1912 . Vorlesugen Über Differentialgleichungen mit Bekannten Infinitesimalen Transformationen , Leipzig : Teubner .
   Google Scholar
• Man , Y. K. 1993 . Computing closed form solutions of first-order ODEs using the Prelle–Singer procedure . J. Symb. Comput. , 16 : 423 – 443 .
   Google Scholar
• Man , Y. K. 1994 . First integrals of autonomous systems of differential equations and the Prelle–Singer procedure . J. Phys. A (Math. Gen.) , 27 : L329 – L332 .
   Google Scholar
• Man , Y. K. and MacCallum , M. A.H. 1996 . A rational approach to the Prelle–Singer algorithm . J. Symb. Comput. , 11 : 1 – 11 .
   Google Scholar
• Olver , P. J. 1993 . Applications of Lie Groups to Differential Equations , New York : Springer–Verleg .
   Google Scholar
• Polyanin , A. D. and Zaitsev , V. F. 1995 . Handbook of Exact Solutions for Ordinary Differential Equations , London : CRC Press .
   Google Scholar
• Polyanin , A. D. , Zaitsev , V. F. and Mousssiaux , A. 2002 . Handbook of First Order Partial Differential Equations , London : Taylor and Francis .
   Google Scholar
• Prelle , M. and Singer , M. 1983 . Elementary first integrals of differential equations . Trans. Am. Math. Soc. , 279 : 215 – 229 .
   Web of Science ®Google Scholar
• Senthil Velan , M. and Lakshmanan , M. 1995 . Lie symmetries and infinite-dimensional Lie algebras of certain nonlinear dissipative systems . J. Phys. A (Math. Gen.) , 28 : 1929 – 1942 .
   Google Scholar
• Sherring , J. and Prince , G. 1992 . Geometric aspects of reduction of order . Trans. Amer. Math. Soc. , 334 : 433 – 453 .
   Web of Science ®Google Scholar
• Shtokhamer , R. 1988 . “ Solving first-order differential equations using the Prelle–Singer algorithm ” . Center for Mathematical Computation, University of Delaware . Tech. Rep. 88–09
   Google Scholar
• Singer , M. 1990 . Formal solutions of differential equations . J. Symb. Comput. , 10 : 59 – 94 .
   Google Scholar
• Singer , M. 1992 . Liouvillian first integrals of differential equations . Trans. Am. Math. Soc. , 333 : 673 – 688 .
   Web of Science ®Google Scholar
• Stephani , H. 1989 . Differential Equations: Their Solutions Using Symmetries , New York : Cambridge University Press .
   Google Scholar