Abel equation of the first kind

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

[math]\displaystyle{ y'=f_3(x)y^3+f_2(x)y^2+f_1(x)y+f_0(x) \, }[/math]

where [math]\displaystyle{ f_3(x)\neq 0 }[/math]. If [math]\displaystyle{ f_3(x)=0 }[/math] and [math]\displaystyle{ f_0(x)=0 }[/math], or [math]\displaystyle{ f_2(x)=0 }[/math] and [math]\displaystyle{ f_0(x)=0 }[/math], the equation reduces to a Bernoulli equation, while if [math]\displaystyle{ f_3(x) = 0 }[/math] the equation reduces to a Riccati equation.

Properties

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

[math]\displaystyle{ uu'=-f_0(x)u^3-f_1(x)u^2-f_2(x)u-f_3(x). \, }[/math]

The substitution

[math]\displaystyle{ \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} }[/math]

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

[math]\displaystyle{ u'=u^3+\phi(\xi). \, }[/math]

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. https://link.springer.com/article/10.1023/A:1014925032022. . (Old link: On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term )
• 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]