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Math Help - [SOLVED] DEq; getting weird answer

  1. #1
    Member pberardi's Avatar
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    [SOLVED] DEq; getting weird answer

    Hey all,
    Solve: x(dy/dx) = y + 2xe^(-y/x)
    let v = y/x
    v + x(dv/dx) = v + 2e^(-v) two or three steps later gives:
    e^vdv = 2(dx)/x integrate and then take ln gives this
    v = ln|2ln|x| + C| this is what i am really wondering about
    y/x = ln|2ln|x| + C|
    y = xln|2ln|x| +C| this is really weird looking, did i do something wrong? Ive done it like five times.

    Thanks,
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  2. #2
    Junior Member
    Joined
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    Australia
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    Quote Originally Posted by pberardi View Post
    Hey all,
    Solve: x(dy/dx) = y + 2xe^(-y/x)
    let v = y/x
    v + x(dv/dx) = v + 2e^(-v) two or three steps later gives:
    e^vdv = 2(dx)/x integrate and then take ln gives this
    v = ln|2ln|x| + C| this is what i am really wondering about
    y/x = ln|2ln|x| + C|
    y = xln|2ln|x| +C| this is really weird looking, did i do something wrong? Ive done it like five times.

    Thanks,
    Looks good. The inner absolute value signs are necessary but can be avoided by expressing this bit as \ln x^2. The other absolute value signs should not be there and really the term (C + \ln x^2) should be positive if we are assuming your answer is real valued. But all your steps are right.

    So assuming we can take x,y to be real, I would write the answer

    y = x \ln\left(C + \ln x^2 \right).

    HTH
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  3. #3
    Member pberardi's Avatar
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    Thanks.
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