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?

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- Dec 29th 2010, 06:34 PMGlitchLinear first order ODE question
**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? - Dec 29th 2010, 06:39 PMpickslides
add $\displaystyle xy$ to both sides, then divide $\displaystyle x^2$ to both sides, does it separate?

- Dec 29th 2010, 06:41 PMProve It
- Dec 29th 2010, 06:44 PMGlitch
Thanks guys, I'll give it another shot.

- Dec 29th 2010, 06:48 PMGlitch
- Dec 29th 2010, 06:51 PMProve It
You can if you assume that $\displaystyle \displaystyle x \neq 0$. It's a standard technique.

- Dec 29th 2010, 06:53 PMGlitch
Ahh, ok. Thanks.

- Dec 29th 2010, 06:56 PMProve It
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.