Classifying a First Order Differential Equation: Separable or Not?

[JinM]

Homework Statement

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

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?

 

[gabbagabbahey]

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

Try the substitution u(x)=x/y instead.

 

[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.

 

[gabbagabbahey]

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