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Math Help - 1st order DE

  1. #1
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    1st order DE

    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?

    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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  2. #2
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    Quote Originally Posted by nocturnal View Post
    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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