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Abel Equation of the First Kind

Niels Abel.

The Abel equation of the first kind, named after Niels Abel, is any ordinary differential equation that is cubic in the unknown function. In other words, it is an equation of the form

y′=f3(x)y3+f2(x)y2+f1(x)y+f0(x)y’=f_3\left(x\right)y^3+f_2\left(x\right)y^2+f_1\left(x\right)y+f_0\left(x\right)

where

f3(x)≠0f_3\left(x\right)\neq 0

If

f3(x)=0 and f0(x)=0, or f2(x)=0 and f0(x)=0f_3\left(x\right)=0 \text{ and } f_0\left(x\right)=0 \text{, or } f_2\left(x\right)=0 \text{ and } f_0\left(x\right)=0

the equation reduces to a Bernoulli equation, while if

f3(x)=0f_3\left(x\right)=0

the equation reduces to a Riccati equation.

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