# Let $y+xy'=x$, $y(0)=-1$. Find $\lim_{x \to \infty}y(x)$

Let $y+xy'=x$, $y(0)=-1$.

Find $\;\lim_{x \to \infty}y(x).$

I have tried to separate $x$ and $y$. $$\frac{y}{1-y'}=x.$$ I don't know how to integrate the left hand side of the function.

Could any one give a fast solution to the problem?

Thank you!

The initial condition might be wrong. Could anyone give a general solution to the problem or you can set a reasonable initial value.

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One way to find the limit is to solve the DE. But since it does not ask that, maybe there is an easier way... But (as noted by Norbert) since no solution satisfies that initial condition, it would be only a hypothetical determination of the limit... – GEdgar Nov 3 '12 at 15:56

Hint: To solve this differential equation note that $$y+xy'=(xy)'$$ P.S. Your requirement for initail conditions is impossible to satisfy. Just substitute $x=0$ to see that $$y(0)=y(0)+0y'(0)=0$$
 +1 Yes. I checked it but didn't want to put a comment. – Babak S. Nov 3 '12 at 16:14 $(xy)'=x'y+xy'x'=y+xy'"$? What's wrong? Could you please explain this step? Thanks : ) – Joe Nov 3 '12 at 17:11 @BabakSorouh I know he is right and I just want to figure out how it comes. Thanks : ) – Joe Nov 3 '12 at 17:33 @BabakSorouh I suppose y is a function about x and then apply implicit rule. Is this step wrong? – Joe Nov 3 '12 at 17:44 @PENGTENG: Sorry brother. You have been right. I didn't see your second terms in second comment above. You are right. But I think the initial conditions is somehow wrong. Can you check it? – Babak S. Nov 3 '12 at 17:47