More general solutions of Abel ODE, second type ?

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The discussion focuses on the general solutions of the second type of Abel Ordinary Differential Equation (ODE), specifically the equation (y(x)+g(x)) y'(x) = f2(x) y(x)^2 + f1(x) y(x) + f0(x). The user is investigating cases where g(x) = 0 and the functions f1, f2, and f0 are polynomials of at most second order divided by x. The conversation also touches on the relationship between Abel equations and Riccati equations, noting that Riccati is a specific case of the first type of Abel equation, which in turn is a specific case of the second type. The relevance of Abel equations in transforming standard equations in mathematical physics is highlighted.

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more general solutions of Abel ODE, second type ??

Is there any progress in solving Abel ODE of second type:

(y(x)+g(x)) y'(x) = f2(x) y(x)^2+f1(x) y(x)+f0(x)

where the unknown function is y(x) and the other functions are coefficient functions.

In the case I am trying to solve, g(x)=0, and the f's are polynomials of at most second order divided by x.

Any reference to the cutting edge solutions of this type of equation appreciated.
 
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The RHS of the equation look familier like Riccati equation.

Just curious, what is Abel equation of the first type? Is there any relation between Abel and Riccati equations?

What's is interesting about Abel equation?
 


Ricatti is particular case of Abel first kind which is particular case of Abel second kind.

The only interesting thing about Abel is that I have to solve a particular instance of it for my research :) Also many standard equations in mathematical physics are transformable to Abel.
 
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