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