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Math Help - Differential Equations

  1. #1
    Junior Member
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    Differential Equations

    Please help me to solve these equations:

    y'+\frac{y}{x}=(xy)^{2}
    xy''-6y'=12x^{5}
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  2. #2
    A Plied Mathematician
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    Well, the second equation is a Cauchy-Euler type if you multiply through by x. Or, since there's no y term, you can reduce the order.
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  3. #3
    MHF Contributor chisigma's Avatar
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    The DE...

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

    ... is of the type analysed in XVII° century by the Swiss mathematician Jacob Bernoulli. Deviding both terms of (1) by y^{2} You obtain first...

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

    ... and then setting \frac{1}{y}= z You obtain...

    \displaystyle z^{'} = \frac{z}{x} - x^{2} (3)

    ... which is linear in z...

    Kind regards

    \chi \sigma
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