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