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

Question

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!

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.

Babak S. · Accepted Answer · 2012-11-05 12:29:59Z

up vote 1 down vote accepted

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?

answered Nov 5 '12 at 12:29

Babak S.
27.9k21249

	Thanks. Could you please provide details and final solution so that I can check my answer and others can learn from your example.Also,I will accept your answer. Thanks. – Joe Nov 8 '12 at 17:08
	Nicely done, as usual! – amWhy Apr 3 at 0:59

Babak S. · Answer 2 · 2012-11-05 12:34:35Z

up vote 1 down vote

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

edited Nov 5 '12 at 12:34

Babak S.
27.9k21249

answered Nov 5 '12 at 12:30

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

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