Non-linear first order differential equation not separable Can you please help with this non-linear first order DE
$$2y\frac{dy}{dx} + 2y =\frac{x^2}{2} + x$$
Rearranged as below does not seem possible to separate variables so what now?
$$\frac{dy}{dx} = \frac{x^2 + 2x}{4y} - 1$$
Regards - Ian
 A: There is no closed form solution, but as the comments mention, we can resort to direction fields to study the behavior of this system.
We see that there are some points interest, that are called fixed points, that is where the derivative is fixed at some point (for example, solve the RHS of your DEQ by setting it equal to zero).
A direction field plot produces:

If we add initial conditions, we see some strange behaviors because $y = 0$ is problematic, for example. Here are some initial conditions to clearly see how solutions behave (the phase lines tell you the direction field that the derivative takes).

A: You can have the following closed form solution but I do not know if it is useful for your purpose

$$ x  =\sqrt [3]{-6\,y-6\,c-1+2\,\sqrt {-16\,{y}^{3}+9\,{
c}^{2}+18\,cy-3\,{y}^{2}+3\,c}}-{\frac {-4\,y-1}{\sqrt [3]{-6\,y-6\,c-
1+2\,\sqrt {-16\,{y}^{3}+9\,{c}^{2}+18\,cy-3\,{y}^{2}+3\,c}}}}-1.$$ 

On the other hand, you can have an approximate solution using the power series techniques

$$ y(x) = y_ 0  -x+\frac{1}{4y_0}\, {x}^{2}+\frac{1}{12}\,{\frac {y_ 0 +2}{
 y_ 0^2}}{x}^{3}+O \left( {x}^{4}\right).$$

A: Hint:
$2y\dfrac{dy}{dx}+2y=\dfrac{x^2}{2}+x$
$2y\left(\dfrac{dy}{dx}+1\right)=\dfrac{x^2+2x}{2}$
$\dfrac{dy}{dx}=\dfrac{x(x+2)}{4y}-1$
Let $u=\dfrac{y}{x+2}$ ,
Then $y=(x+2)u$
$\dfrac{dy}{dx}=(x+2)\dfrac{du}{dx}+u$
$\therefore(x+2)\dfrac{du}{dx}+u=\dfrac{x}{4u}-1$
