# Another 1st order diffy eq.

1. Oct 11, 2008

### JinM

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

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

3. The attempt at a solution

If you divide both sides by x, and substitute u = y/x and y' = u'x + u, we get

$$u'x =ue^{\frac{1}{u}} - u - 1$$.

This is seperable, but how the heck do you integrate the RHS? Or could we just say, like what we do in linear DE's, that

(ux)' = e^(1/u) - 1, and then integrate both sides? Although unusual, is that correct?

Last edited: Oct 11, 2008
2. Oct 11, 2008

### gabbagabbahey

How would you integrate e^(1/u)-1 dx without knowing u(x)?!

3. Oct 11, 2008

### JinM

Still nonelementary. I think there must be a typo in the book because an integral like that is unusual for this class. Heck, maybe it isn't a typo at all because the book is just asking us to classify said differential equation.

4. Oct 11, 2008

### gabbagabbahey

Hmmm. yes not easily integrated, but since all you're asked to do is classify the DE, its clearly separable.