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Math Help - First-order linear DE--help w/ integrating factor

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    First-order linear DE--help w/ integrating factor

    (xy'-1)\ln x=2y
    y'x\ln x -\ln x=2y
    y'x\ln x -2y=\ln x
    y'-\frac{2y}{x\ln x}=\frac{1}{x}

    so P(x) = e^{\int x\ln x~dx}

    Help with that integration, please? Should I try substitution, or should I go with integration by parts?
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    Re: First-order linear DE--help w/ integrating factor

    Quote Originally Posted by phys251 View Post
    (xy'-1)\ln x=2y
    y'x\ln x -\ln x=2y
    y'x\ln x -2y=\ln x
    y'-\frac{2y}{x\ln x}=\frac{1}{x}

    so P(x) = e^{\int x\ln x~dx}

    Help with that integration, please? Should I try substitution, or should I go with integration by parts?
    First of all, your integrating factor is actually \displaystyle \begin{align*} e^{\int{-\frac{2}{x\ln{x}}\,dx}} \end{align*}. You can evaluate this integral using the substitution \displaystyle \begin{align*} u = \ln{x} \implies du = \frac{1}{x}\,dx \end{align*}.
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    Re: First-order linear DE--help w/ integrating factor

    Ah, that did it. Thanks.
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