Consider the differential equation
$$ y' = a_0(x) + a_1(x)y + a_2(x)\frac{1}{y}$$
I am attempting to find the general solution to this. One thing I can note is that the entire equation can be rewritten as
$$ y' = \frac{a_2(x) + a_0(x)y + a_1(x)y^2}{y} $$
Thus allowing us to state
$$ y y' = a_2(x) + a_0(x)y + a_1(x)y^2$$
I have no idea how to progress correctly from here.
By General Solution, I mean to ask if this can be re-written as a linear ODE.
This belongs to an Abel equation of the second kind.
In fact all Abel equation of the second kind can be transformed into Abel equation of the first kind.
Let $y=\dfrac{1}{u}$ ,
Then $y'=-\dfrac{u'}{u^2}$
$\therefore-\dfrac{u'}{u^2}=a_0(x)+\dfrac{a_1(x)}{u}+a_2(x)u$
$u'=-a_2(x)u^3-a_0(x)u^2-a_1(x)u$
Please follow the method in http://www.hindawi.com/journals/ijmms/2011/387429/#sec2