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 $f_{3}(x)\neq 0$.

Properties

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.

Solution

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

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

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 in an implicit form.

References

• 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.
• Mancas, Stefan C.; Rosu, Haret C. (2013). "Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations". Physics Letters A. 377: 1434–1438. arXiv:1212.3636. doi:10.1016/j.physleta.2013.04.024.