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Another 1st order diffy eq.

  1. Oct 11, 2008 #1
    1. The problem statement, all variables and given/known data

    [tex]x\frac{dy}{dx} = ye^{\frac{x}{y}} - x[/tex]

    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

    [tex]u'x =ue^{\frac{1}{u}} - u - 1[/tex].

    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. jcsd
  3. Oct 11, 2008 #2


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

    Try the substitution u(x)=x/y instead.
  4. Oct 11, 2008 #3
    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.
  5. Oct 11, 2008 #4


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