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Help with first order, Bernoulli ODE

  1. Feb 11, 2008 #1
    Help with first order, "Bernoulli" ODE

    We just covered:
    -First order linear ordinary differential equations
    -Bernoulli Equations
    -Simple substitutions.

    This problem was assigned. Its supposedly a Bernoulli equation with respect to y, but I can't figure it out...


    When I solve for dx/dy, I get dx/dy = x^3 -y/x, which is not a Bernoulli equation because of the factor of 1/x, and not x. Help?
  2. jcsd
  3. Feb 11, 2008 #2
  4. Feb 11, 2008 #3
    I didn't get an answer at all. My problem is that I could not convert it into a form which I can use Bernoulli's substitutions on it. I tried finding an explicit for both y(x) and x(y).
  5. Feb 11, 2008 #4
    I doesn't look like Bernoulli's equation but I wonder if you can use similar techniques. Is there a change of variables you can apply to put it into a form that you know how to solve?
  6. Feb 12, 2008 #5
    A basic u = x^4-y substitution did not yield a linear (or a Bernoulli) differential equation =( I'm stumped. Does anybody mind steering me in the right direction?
    Last edited: Feb 12, 2008
  7. Feb 13, 2008 #6
    Can you try forming the equation for the variable x?
  8. Feb 14, 2008 #7
    The ODE [itex](x^4-y(x))\,y'(x)=x[/itex] is not a Bernoulli equation and furthermore is not a simple one :smile:
    You can transformed it into a Riccati one by the transformation


    which makes the ODE


    Now letting
    we arrive to


    which is the definition of the Airy function.
  9. Feb 14, 2008 #8
    Ya..As it turns out, our teacher gave us the wrong problem.......
    Thanks though! I really appreciate it!

    (Its kind of interesting how tiny changes in the terms creates such a huge difference in difficulty)
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