Solving a specific case of Abel's differential equation of the second kind I'd really like to solve this equation:
$$y \frac{dy}{dx} = -y - 2x^2(1-x)$$
This equation cropped up in my research, so I don't know whether a solution is even possible.  I tried solving it, but I got nowhere.
Please let me know if you:
1) Know how to solve it
2) Can suggest a possible method for solving it
3) Know that it is not possible to solve it
Thank you!

Edit:  In the process of formulating this equation originally, I missed a sign (embarrasing, so sorry!).   I've changed the sign that I missed and now a particular solution seems to be just:
$$y(x) = x^2 - x$$
So, from my point of view, this (new) problem is now partly solved.   If anyone wants to see the incorrect equation for fun, I'm reprinting it here:
$$y \frac{dy}{dx} = y - 2x^2(1-x)$$
As @JJacquelin points out in the comments, a particular solution to this is
$$y(x) = x- x^2$$
Again, my apologies for initially posting the incorrect equation.

Edit 2
As it turns out, the particular solution satisfies my initial condition $y(0)=0$, so it solves this problem for my current purposes.  However, I am still curious as to whether a general solution is possible, so I will leave the question open.  I also wonder whether this particular solution is the only one to satisfy my initial condition, or whether there are others.
 A: This is not a complete answer to the question, but it's too long for a comment, and I just want to note a typical approach for solving ODE's which can not be used in this case. This doesn't mean that there doesn't exist any method to solve it -- this just means that this common method does not work.
Rewriting the equation as $$y \frac{dy}{dx} + y = -2x^2(1-x) $$ we see that this is not a linear ODE.
If we tried to write the LHS as a differential operator, we would see that it has the form
$$ M  y , \quad \text{where} \quad M = y D + y$$
We see that the coefficient of the first derivative $D$ is not a function of $x$, but a function of $y$. Therefore the differential operator $M$ for this equation is not a linear operator, and the standard methods which apply to linear ODE's do not apply.
Therefore, if you want to find a solution to the equation, it will have to be a method for solving (certain classes of) nonlinear ODE's.
I know this does not solve your problem, but hopefully will prevent you from pursuing unnecessary dead-ends.
