Abel equation of the first kind

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) \,

where f_3(x)\neq 0. If f_3(x)=0 and f_0(x)=0, or f_2(x)=0 and f_0(x)=0, the equation reduces to a Bernoulli equation, while if f_3(x) = 0 the equation reduces to a Riccati equation.

Properties

The substitution y=\dfrac{1}{u} brings the Abel equation of the first kind to the "Abel equation of the second kind" of the form

uu'=-f_0(x)u^3-f_1(x)u^2-f_2(x)u-f_3(x). \,

The substitution

\begin{align} \xi & = \int f_3(x)E^2~dx, \\[6pt] u & = \left(y+\dfrac{f_2(x)}{3f_3(x)}\right)E^{-1}, \\[6pt] E & = \exp\left(\int\left(f_1(x)-\frac{f_2^2(x)}{3f_3(x)}\right)~dx\right) \end{align}

brings the Abel equation of the first kind to the canonical form

u'=u^3+\phi(\xi). \,

Dimitrios E. Panayotounakos and Theodoros I. Zarmpoutis discovered an analytic method to solve the above equation generally.[1]

Notes

  1. ↑ E. Panayotounakos, Dimitrios; Theodoros I. Zarmpoutis (29 August 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 Volume 2011 (2011) 2011: 13. doi:10.1155/2011/387429. PMID 387429.

References