1st order DE

Quote:

Originally Posted by nocturnal

Solve the following differential equation:

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

Well, an appropriate substitution would be $y = ux$ , so that $y' = u'x + u.$

After dividing both sides by x, and substituting the above, we get

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

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

$(ux)' = ue^{\frac{1}{u}} - 1$ , and then integrate both sides? Mr F says: No. The integration would have to be wrt x, but the right hand side is a function of u. You have to seperate and integrate. The integral wrt u is certainly non-elementary. I'd re-check the question for a typo.

Although unusual (and gives what looks like a simpler integral to solve), I don't think $ue^{\frac{1}{u}}$ can be integrated using elementary functions.

Can anyone see a way out?

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