More general solutions of Abel ODE, second type ?

In summary, there has been progress in finding more general solutions for Abel ODE of the second type, where the unknown function is y(x) and the other functions are coefficient functions. There are polynomial functions of at most second order divided by x. The first type of Abel equation is related to Riccati equation, and the second type is a particular case of the first type. Abel equations are interesting because they can be transformed into other standard equations in mathematical physics.
  • #1
smallphi
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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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  • #2


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?
 
  • #3


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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FAQ: More general solutions of Abel ODE, second type ?

What is an Abel ODE of the second type?

An Abel Ordinary Differential Equation (ODE) of the second type is a type of mathematical equation that involves a dependent variable, an independent variable, and their respective derivatives. It is named after mathematician Niels Henrik Abel and is characterized by the presence of an integral term.

What are the general solutions of Abel ODE, second type?

The general solutions of an Abel ODE of the second type are equations that satisfy the original differential equation for all possible values of the independent variable. They are expressed in terms of the dependent variable and its derivatives, as well as an arbitrary constant or constants.

How do you solve a second-order Abel ODE?

To solve a second-order Abel ODE, you can use various methods such as separation of variables, substitution, or the Laplace transform. However, for more complicated equations, numerical methods or approximation techniques may be necessary.

What is the significance of Abel ODE, second type in science?

Abel ODEs of the second type have applications in various fields of science, including physics, engineering, and biology. They are useful for modeling systems with changing rates, such as chemical reactions, population growth, and electrical circuits.

Are there any real-world examples of Abel ODE, second type?

Yes, there are many real-world examples of Abel ODEs of the second type. Some common examples include the Brachistochrone problem in physics, the Riccati equation in control theory, and the Lotka-Volterra equations in population dynamics.

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