In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an Abel equation of the first kind, named after
Niels Henrik Abel, is any
ordinary differential equation that is
cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system w ...
in the unknown function. In other words, it is an equation of the form
:
where
. If
and
, or
and
, the equation reduces to a
Bernoulli equation, while if
the equation reduces to a
Riccati equation.
Properties
The substitution
brings the Abel equation of the first kind to the "
Abel equation of the second kind
Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepher ...
" 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.
Notes
References
*. (Old link
''On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term''{Dead link, date=June 2020 , bot=InternetArchiveBot , fix-attempted=yes )
''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.
rXiv.org:1212.3636v3
Ordinary differential equations