I have a question regarding "Endless Transformation. I'm actually working on Riccati Equation and I found a Post here on MO (**[Looking for the solution of first order non-linear differential equation ($y ′+y^{2}=f(x)$) without knowing a particular solution.](https://mathoverflow.net/questions/87041/looking-for-the-solution-of-first-order-non-linear-differential-equation-y-y)**), wherein the equation was solved using Endless transformation. I'm really curious about that method but I can't find any references wherein Endless Transform was elaborated. 


**Is "Endless Transform" really an existing Method?** I can't search it on Google.
Does it have another name?
Who established the Method?


I really need your help.
Thanks!



Endless Transformation in finding Particular Solution of Riccati Equation
It is too long for a comment , Thus I wrote an answer


I am the person who wrote the transform that you mentioned in your question. I did not see anywhere the term of endless transform, I just noticed if we apply the variable transform 
$$y=\frac{f(x)}{\frac{f'(x)}{2f(x)}+y\_1} $$ on Riccati Differential Equation $y'+y^{2}=f(x)$,  

We get same type Riccati Differantial Equation after the transform.
$$y\_1 '+y\_1^{2}=f\_1(x)=f(x)+(\frac{-f'(x)}{2f(x)})^{2}+(\frac{-f'(x)}{2f(x)})'$$


Then, if we apply the transform $$y\_1=\frac{f\_1(x)}{\frac{f\_1'(x)}{2f\_1(x)}+y\_2} $$ on $y\_1 '+y\_1^{2}=f\_1(x)$ , we get the same type equation again $$y\_2 '+y\_2^{2}=f\_2(x)=f\_1(x)+(\frac{-f\_1'(x)}{2f\_1(x)})^{2}+(\frac{-f\_1'(x)}{2f\_1(x)})'$$ 
It never ends , we always get the same type equation ($y'+y^{2}=f(x)$) , Thus I wrote it as endless transform.


Finally, the particular solution can be written in infinite terms as


$$y\_p(x)=\frac{f(x)}{\frac{f'(x)}{2f(x)}+\frac{f\_1(x)}{\frac{f\_1'(x)}{2f\_1(x)}+\frac{f\_2(x)}{\frac{f\_2'(x)}{2f\_2(x)}+.....}} } $$


Where $$f\_{n+1}(x)=f\_n(x)+(\frac{-f\_n'(x)}{2f\_n(x)})^{2}+(\frac{-f\_n'(x)}{2f\_n(x)})'$$
and $$f\_0(x)=f(x)$$


I do not know the name of this transform in literature, So I named it as endless transform. Maybe someone else can help to find what the name of the transform is in literature.


I hope it helps to enlighten you


