# Abel Equations

An Abel ODE is a first-order equation of the form

This equation arose in the context of the studies of Niels Henrik Abel on the theory of elliptic functions, and represents a natural generalization of the Riccati equation.

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.

Here is the construction of a particular invariant with value 0 and the solution of the corresponding Abel ODE.

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Another important class of integrable Abel ODEs are those that can be reduced to inverse linear first-order ODEs using a nonlinear coordinate transformation.

This Abel ODE is solved by transforming it to an inverse linear first-order ODE. The

ExpIntegralEi term in the solution to this equation comes from solving the linear ODE.

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.

This Abel ODE is solved by reducing it to an inverse Riccati equation.

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This verifies the solution.

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The Abel ODEs considered so far are said to be of the *first kind.* Abel ODEs of the *second kind* are given by the following general formula.

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.

Here is the solution for an Abel ODE of the second kind.

This verifies the solution.

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