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Abel equation of the first kind

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

where f 3 ( x ) ≠ 0 {\displaystyle f_{3}(x)\neq 0} .

Properties

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

Solution

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

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

The substitution

ξ = ∫ f 3 ( x ) E 2   d x , u = ( y + f 2 ( x ) 3 f 3 ( x ) ) E − 1 , E = exp ⁡ ( ∫ ( f 1 ( x ) − f 2 2 ( x ) 3 f 3 ( x ) )   d x ) {\displaystyle {\begin{aligned}\xi &=\int f_{3}(x)E^{2}~dx,\\u&=\left(y+{\dfrac {f_{2}(x)}{3f_{3}(x)}}\right)E^{-1},\\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 + ϕ ( ξ ) . {\displaystyle 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.

Notes

  1. Panayotounakos, Dimitrios E.; Zarmpoutis, Theodoros I. (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. 2011. Hindawi Publishing Corporation: 1–13. doi:10.1155/2011/387429.

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.
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