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Math Help - First-order nonlinear ordinary differential equations

  1. #1
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    First-order nonlinear ordinary differential equations

    I'm having some trouble with these, could you lend me some help?

    This one for example:
    y = x.y' + y'.ln(y')

    I tried p = y' but all i got was this
    dx + (x+ln(p))dp = 0
    and then I finish with an impossible integration
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: First-order nonlinear ordinary differential equations

    Quote Originally Posted by mafra View Post
    I'm having some trouble with these, could you lend me some help?

    This one for example:
    y = x.y' + y'.ln(y')

    I tried p = y' but all i got was this
    dx + (x+ln(p))dp = 0
    and then I finish with an impossible integration
    The DE requires a quite particular approach. We have...

    y=x\ y^{'} + y^{'}\ \ln y^{'} (1)

    Now if we differentiate both terms of (1) we obtain...

    y^{'}= y^{'} + (x+1+\ln y^{'})\ y^{''} (2)

    ... and the (2) is satisfied for...

    y^{''}=0 (3)

    ...or...

    x+\ln y^{'}=-1 (4)

    Thesystem of (1) and (3) has solution...

    y=c\ x + c\ \ln c (5)

    ... and that is the general solution of (1). The system of (1) and (4) has as solution a 'particular solution' of (1) and it requires an easy procedure...

    Kind regards

    \chi \sigma
    Last edited by chisigma; September 11th 2011 at 10:34 PM. Reason: error in (2)...
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  3. #3
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    Re: First-order nonlinear ordinary differential equations

    Thanks, but where's the term for y'(lny')'?

    I think I got the singular solution too; -e^-x
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  4. #4
    MHF Contributor chisigma's Avatar
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    Re: First-order nonlinear ordinary differential equations

    Quote Originally Posted by mafra View Post
    Thanks, but where's the term for y'(lny')'?

    I think I got the singular solution too; -e^-x
    Your replay has been useful because I discovered an error. It is...

    \frac{d}{dx} (y^{'}\ \ln y^{'}) = (1+\ln y^{'})\ y^{''} (1)

    ... so that the second equation is...

    x+\ln y^{'}=-1 (2)

    Kind regards

    \chi \sigma
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