Abel equation of the first kind

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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

where . If and , or and , the equation reduces to a Bernoulli equation, while if the equation reduces to a Riccati equation.

Properties[edit]

The substitution brings the Abel equation of the first kind to the "Abel equation of the second kind" of the form

The substitution

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

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

Notes[edit]

  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. Hindawi Publishing Corporation. 2011: 1–13. doi:10.1155/2011/387429. 

References[edit]