Geometric Properties of Abel Differential Equations of the First and Second Kind
DOI:
https://doi.org/10.29020/nybg.ejpam.v1i2.7800Keywords:
Abel equation, first kind, projective, transformation, solution space, integrabilityAbstract
A number of different procedures and techniques are used to study the Abel and some related equations of the first and second kind. The Abel equation of the first kind is related to a non-homogeneous second order Li\'enard equation. These equations are equivalent as far as algebraic and Liouvillian integrability go. Any Liouvillian solution to a Riccati equation induces a Liouvillian solution for the second order associated second order linear equation and conversely. Under suitable conditions on the coefficients which appear in the Abel equation, a Darboux first integral can be produced for the Abel equation of the first kind. Some results related to algebraic solutions to one version of the equation are developed at the end.
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Copyright (c) 2026 Paul Bracken
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