There is a general solution from the Polyanin textbook for the equation,
$y\cdot\frac{dy}{dx}-y=Ax+B$
The solution in parametric form is
$x = C \cdot e^{-\int \frac {t \cdot dt}{t^2-t-A}} $ and $y = C \cdot t \cdot e^{-\int \frac {t \cdot dt}{t^2-t-A}} $
I tried to derive the solution without success. How it was derived?
The equation can be written as $$ \dfrac{dy}{dx} = \frac{y + A x + B}{y}$$ A linear change of independent variable $X = A x + B$ makes this into a homogeneous equation $$ \dfrac{dy}{dX} = \frac{y+X}{Ay} $$ Now if $y = X t$, this becomes $$ \dfrac{dt}{dX} = \frac{1 + t - A t^2}{A X t}$$ Now consider $X$ as the independent variable and $t$ as dependent: $$ \dfrac{dX}{dt} = \frac{AX t}{1 + t - A t^2} $$ This is a homogeneous linear differential equation. Use the standard formula.