Existence of closed form solution of the below DE I have a problem finding an analytical closed form solution of the following DE. 
$ \displaystyle \left( \frac{y(x)+2}{3} \right) \frac{dy}{dx} + y(x) +\frac{1}{x} \left( a y(x)^2 + b y(x) -c \right) =0. \\
\{a,b,c\} \in \mathbb{R}, \qquad \{a,b,c\}>0.
$
I encountered this differential equation in my research in fluid dynamics. I failed to reduce it to some standard form of DE for which solutions are possible. Mathematica and Maple are unable to find analytical solutions for this. 
Can anyone point out any method that can be used to know if a solution for this DE exists?
And if a solution exists, can anyone point out a method to solve the DE?     
 A: Hint:
Approach $1$:
This belongs to an Abel equation of the second kind.
Let $u=y+2$ ,
Then $\dfrac{du}{dx}=\dfrac{dy}{dx}$
$\therefore\dfrac{u}{3}\dfrac{du}{dx}+u-2+\dfrac{1}{x}(a(u-2)^2+b(u-2)-c)=0$
$\dfrac{u}{3}\dfrac{du}{dx}+u-2+\dfrac{1}{x}(au^2-(4a-b)u+4a-2b-c)=0$
$u\dfrac{du}{dx}=-\dfrac{3au^2}{x}-3\left(1-\dfrac{4a-b}{x}\right)u+6-\dfrac{3(4a-2b-c)}{x}$
Let $u=x^{-3a}v$ ,
Then $\dfrac{du}{dx}=x^{-3a}\dfrac{dv}{dx}-3ax^{-3a-1}v$
$\therefore x^{-3a}v\left(x^{-3a}\dfrac{dv}{dx}-3ax^{-3a-1}v\right)=-\dfrac{3ax^{-6a}v^2}{x}-3\left(1-\dfrac{4a-b}{x}\right)x^{-3a}v+6-\dfrac{3(4a-2b-c)}{x}$
$x^{-6a}v\dfrac{dv}{dx}-3ax^{-6a-1}v^2=-3ax^{-6a-1}v^2-3\left(1-\dfrac{4a-b}{x}\right)x^{-3a}v+6-\dfrac{3(4a-2b-c)}{x}$
$x^{-6a}v\dfrac{dv}{dx}=-3\left(1-\dfrac{4a-b}{x}\right)x^{-3a}v+6-\dfrac{3(4a-2b-c)}{x}$
$v\dfrac{dv}{dx}=-3(x^{3a}-(4a-b)x^{3a-1})v+6x^{6a}-3(4a-2b-c)x^{6a-1}$
Approach $2$:
$\dfrac{y+2}{3}\dfrac{dy}{dx}+y+\dfrac{ay^2+by-c}{x}=0$
$y+\dfrac{ay^2+by-c}{x}=-\dfrac{y+2}{3}\dfrac{dy}{dx}$
$\left(x+ay+b-\dfrac{c}{y}\right)\dfrac{dx}{dy}=-\left(\dfrac{1}{3}+\dfrac{2}{3y}\right)x$
This belongs to an Abel equation of the second kind.
Let $u=x+ay+b-\dfrac{c}{y}$ ,
Then $x=u-ay-b+\dfrac{c}{y}$
$\dfrac{dx}{dy}=\dfrac{du}{dy}-a-\dfrac{c}{y^2}$
$\therefore u\left(\dfrac{du}{dy}-a-\dfrac{c}{y^2}\right)=-\left(\dfrac{1}{3}+\dfrac{2}{3y}\right)\left(u-ay-b+\dfrac{c}{y}\right)$
$u\dfrac{du}{dy}-\left(a+\dfrac{c}{y^2}\right)u=-\left(\dfrac{1}{3}+\dfrac{2}{3y}\right)u+\dfrac{(y+2)(ay^2+by-c)}{3y^2}$
$u\dfrac{du}{dy}=\dfrac{(3a-1)y^2-2y+3c}{3y^2}u+\dfrac{(y+2)(ay^2+by-c)}{3y^2}$
