# Homework Help: Help with differential equation

1. Feb 26, 2010

### 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:

1. The problem statement, all variables and given/known data
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)

2. Feb 26, 2010

### 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.