The substitution brings the Abel equation of the first kind to the "Abel equation of the second kind" of the form
The substitution
brings the Abel equation of the first kind to the canonical form
Dimitrios E. Panayotounakos and Theodoros I. Zarmpoutis discovered an analytic method to solve the above equation in an implicit form.[1]
NotesEdit
^Panayotounakos, Dimitrios E.; Zarmpoutis, Theodoros I. (2011). "Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)". International Journal of Mathematics and Mathematical Sciences. Hindawi Publishing Corporation. 2011: 1–13. doi:10.1155/2011/387429.
ReferencesEdit
Panayotounakos, D.E.; Panayotounakou, N.D.; Vakakis, A.F.A (2002). "On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term". Nonlinear Dynamics. 28: 1–16. doi:10.1023/A:1014925032022. S2CID 117115358.. (Old link: On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term[permanent dead link])
Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)
Mancas, Stefan C., Rosu, Haret C., Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations. Physics Letters A 377 (2013) 1434–1438. [arXiv.org:1212.3636v3]
October 21, 2023
abel, equation, first, kind, this, article, about, certain, differential, equations, certain, functional, equations, named, after, abel, abel, equation, mathematics, named, after, niels, henrik, abel, ordinary, differential, equation, that, cubic, unknown, fun. This article is about certain differential equations For certain functional equations named after Abel see Abel equation In mathematics an Abel equation of the first kind named after Niels Henrik Abel is any ordinary differential equation that is cubic in the unknown function In other words it is an equation of the form y f 3 x y 3 f 2 x y 2 f 1 x y f 0 x displaystyle y f 3 x y 3 f 2 x y 2 f 1 x y f 0 x where f 3 x 0 displaystyle f 3 x neq 0 If f 3 x 0 displaystyle f 3 x 0 and f 0 x 0 displaystyle f 0 x 0 or f 2 x 0 displaystyle f 2 x 0 and f 0 x 0 displaystyle f 0 x 0 the equation reduces to a Bernoulli equation while if f 3 x 0 displaystyle f 3 x 0 the equation reduces to a Riccati equation Properties EditThe substitution y 1 u displaystyle y dfrac 1 u nbsp brings the Abel equation of the first kind to the Abel equation of the second kind of the form u u f 0 x u 3 f 1 x u 2 f 2 x u f 3 x displaystyle uu f 0 x u 3 f 1 x u 2 f 2 x u f 3 x nbsp The substitution 3 f 3 x E 2 d x u y f 2 x 3 f 3 x E 1 E exp f 1 x f 2 2 x 3 f 3 x d x displaystyle begin aligned xi amp int f 3 x E 2 dx 6pt u amp left y dfrac f 2 x 3f 3 x right E 1 6pt E amp exp left int left f 1 x frac f 2 2 x 3f 3 x right dx right end aligned nbsp brings the Abel equation of the first kind to the canonical form u u 3 ϕ 3 displaystyle u u 3 phi xi nbsp Dimitrios E Panayotounakos and Theodoros I Zarmpoutis discovered an analytic method to solve the above equation in an implicit form 1 Notes Edit Panayotounakos Dimitrios E Zarmpoutis Theodoros I 2011 Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs Abel s Nonlinear ODEs of the First Kind and Relative Degenerate Equations International Journal of Mathematics and Mathematical Sciences Hindawi Publishing Corporation 2011 1 13 doi 10 1155 2011 387429 References EditPanayotounakos D E Panayotounakou N D Vakakis A F A 2002 On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term Nonlinear Dynamics 28 1 16 doi 10 1023 A 1014925032022 S2CID 117115358 Old link On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term permanent dead link Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs Abel s Nonlinear ODEs of the First Kind and Relative Degenerate Equations Mancas Stefan C Rosu Haret C Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations Physics Letters A 377 2013 1434 1438 arXiv org 1212 3636v3 Retrieved from https en wikipedia org w index php title Abel equation of the first kind amp oldid 1152134404, wikipedia, wiki, book, books, library,