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Math Help - epproach on solving this equation

  1. #1
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    epproach on solving this equation

    xy'-2x^{2}\sqrt{y}=4y
    this equation in not saparable so i need to find the integration coefficient?

    is there easier way
    it looks like bernuly but i am not sure
    because of the square root
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  2. #2
    A Plied Mathematician
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    If by "integration coefficient" you mean "integrating factor", that technique only works on first-order linear equations, which this obviously is not. Why not try the Bernoulli approach? Standard form would be

    xy'-2x^{2}\sqrt{y}=4y

    xy'-4y=2x^{2}\sqrt{y}

    y'-\dfrac{4}{x}\,y=2x\sqrt{y}=2xy^{1/2}.

    So the substitution would be what?
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  3. #3
    MHF Contributor chisigma's Avatar
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    Setting z=\sqrt{y} \implies dy=2\ z\ dz You obtain...

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

    ... that is a linear first order ODE and can be 'attacked' in usual way...

    Kind regards

    \chi \sigma
    Last edited by chisigma; October 21st 2010 at 04:32 PM. Reason: error of sigh... sorry!...
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  4. #4
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    thanks
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  5. #5
    A Plied Mathematician
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    You're welcome for my contribution. Incidentally, I believe that the Bernoulli approach would use the same substitution that chisigma proposed.
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