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Math Help - Use of Bernoulli's Equation

  1. #1
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    Use of Bernoulli's Equation

    I have a problem I am working on:

    xy - dy/dx = e^(-3x^2/2) * y^4

    If I solve this using the Bernoulli equation then I have n = 4, 1-n = 3.

    I make the substitution v = y^3 , y = v^1/3

    I'm have a little problem with the next step of finding dx/dy. I'm pretty sure once I find this then I can substitute these values into the DE to get a linear DE that can then be solved.
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  2. #2
    MHF Contributor chisigma's Avatar
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    First we write the DE as...

    y^{'} = xy - e^{-\frac{3}{2} x^{2}} y^{4} \rightarrow \frac{y^{'}}{y^{4}} = \frac{x}{y^{3}} - e^{-\frac{3}{2} x^{2}} (1)

    ... and then we set...

    \frac {1}{y^{4}} = z \rightarrow \frac{y^{'}}{y^{4}}= - \frac{z^{'}}{3} (2)

    Now combining (1) and (2) we arrive to write...

    z^{'} = -3\cdot xz +3\cdot e^{-\frac{3}{2} x^{2}} (3)

    ... which is linear in z ...

    Kind regards

    \chi \sigma
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