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Let $\frac{dy}{dx}-\frac{y}{x}=xe^{-x}$ Find $\lim_{x \to \infty}\frac{y}{x}$

I have tried to separate variables and integral both side of the function. However, it seems impossible.

Could any one give a quick solution?

Thanks!

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    Please don't use the asterisk `*` for multiplication. This symbol stands for convolution in mathematics. If you must add a multiplication symbol, use `\cdot` instead. But in this case, it is not needed. I edited your post accordingly.2012-11-05

2 Answers 2

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The quation will be an exact one if we use the integrating factor $\mu(x)=1/x, x\neq0$. So we have $d(y/x)=\exp(-x)$ and then $(y/x)=-e^{-x}+C$. Now can you find your answer?

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    Nicely done, as usual!2013-04-03
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$y=-xe^{-x} +cx$ integrating factor = $e^{\int\frac{-1}{x}dx}$ proceed as mentioned here http://en.wikipedia.org/wiki/Integrating_factor