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

${\displaystyle y'=f_{3}(x)y^{3}+f_{2}(x)y^{2}+f_{1}(x)y+f_{0}(x)\,}$

where ${\displaystyle f_{3}(x)\neq 0}$. If ${\displaystyle f_{3}(x)=0}$ and ${\displaystyle f_{0}(x)=0}$, or ${\displaystyle f_{2}(x)=0}$ and ${\displaystyle f_{0}(x)=0}$, the equation reduces to a Bernoulli equation, while if ${\displaystyle f_{3}(x)=0}$ the equation reduces to a Riccati equation.

Properties[]

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

${\displaystyle uu'=-f_{0}(x)u^{3}-f_{1}(x)u^{2}-f_{2}(x)u-f_{3}(x).\,}$

The substitution

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

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

${\displaystyle u'=u^{3}+\phi (\xi ).\,}$

and discovered an analytic method to solve the above equation in an implicit form.^[1]

Notes[]

References[]

• 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]