Help with differential equation

[Richirude]

Hello :)

I'm trying to solve an homogeneous equation... but it seems that I'm wrong in some step... or something, because I can't complete this problem, look here is what i got:

Homework Statement

Solve the given homogeneous equation:
$$x\frac{dy}{dx} = ye^\frac{x}{y} - x$$


2. The attempt at a solution

$$x dy = (ye^\frac{x}{y} - x) dx$$

$$x dy + ( -ye^\frac{x}{y} + x) dx$$

Using substitution
$$x = vy$$

$$dx = vdy + ydv$$


$$vy dy + (-ye^v + vy) [ vdy + ydv ] = 0$$


$$(vy + yv^2 - vye^v)dy + (vy^2 - y^2e^v )dv = 0$$


$$(vy + yv^2 - vye^v)dy = (y^2e^v - vy^2 )dv$$


$$\frac{(y)(v + v^2 - ve^v)}{y^2(e^v - v)} dy = dv$$


$$\int \frac{1}{y} dy = \int \frac{(e^v - v)}{(v + v^2 - ve^v)} dv$$

The integral on the left side is easy to solve, but I can't find a way to solve the right side of the equation. Maybe I'm wrong in some earlier step...


Any Suggestions?

Thanks and sorry for my English, I'm still learning it (i'm from venezuela)

 

[LCKurtz]

Your English is quite good; no need to apologize. For what it is worth, I agree with your steps, but I don't see how to work it either. Apparently, neither does Maple.