Understanding Riccati's ODE Variant: A Generalized RODE Explanation

[MathematicalPhysicist]

Any one knows how do you call an equation of the type:
$$y' = q_0(x)+q_1(x) y+...+q_n(x) y^n$$
Maybe generalized RODE?

 

[JJacquelin]

Why do you expect a more specific name for this non-linear ODE ?

 

[MathematicalPhysicist]

Well I didn't find an entry of this equation in wiki, or I didn't search well enough.

So is this its name?

 

[JJacquelin]

May be, there is no analytical solution known in the general case, for n>2.

 

[MathematicalPhysicist]

Do you have any references I can read more on the general case (I mean what has been done already with it)?

 

[MathematicalPhysicist]

Well I can do something like integrate the equation and get:
$$y = \int q_0(x) dx + \int q_1(x) y dx +...\int q_n(x) y^n dx$$

and plug the y in the left in y in the intergals, I would get an infinite sequence, it maybe be good for numerical calculations, but still not analytical.

I can also plug in a power series with powers of x for y.

I guess all of these methods have been tried before.

If I assume that q_i's are analytic and have an overlapping convergence domain, I can show that there should be a solution in a series form.