An Abel ODE is a first-order equation of the form
Associated with any Abel ODE is a sequence of expressions that is built from the coefficients of the equation and invariant under certain coordinate transformations of the independent variable and the dependent variable. These invariants characterize each equation and can be used for identifying integrable classes of Abel ODEs. In particular, Abel ODEs with zero or constant invariants can be integrated easily and constitute an important integrable class of these equations.
Another important class of integrable Abel ODEs consists of those that can be transformed to an inverse Riccati equation. Since Riccati equations can be transformed to second-order linear ODEs, the solutions for this class are usually given in terms of special functions such as AiryAi and BesselJ.
An Abel ODE of the second kind can be converted to an equation of the first kind with a coordinate transformation. Thus, the solution methods for this kind of Abel ODE are identical to the methods for equations of the first kind.