The question:
$\displaystyle x^2 \frac{dy}{dx} -xy = y$
I'm not sure where to start. I tried getting it in a form where I can calculate the integrating factor, but nothing I do seems to work. Any suggestions?
I suppose that you could always write it like this...
$\displaystyle \displaystyle x^2\,\frac{dy}{dx} - x\,y - y = 0$
$\displaystyle \displaystyle x^2\left(\frac{dy}{dx} - x^{-1}y - x^{-2}y\right) = 0$
$\displaystyle \displaystyle x^2\left[\frac{dy}{dx} - (x^{-1} + x^{-2})\,y\right] = 0 $.
You can still use the Integrating Factor Method and then solve the equation using NFL.