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.