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First Order ODE

  1. Jul 21, 2014 #1
    1. The problem statement, all variables and given/known data

    dy/dx = x/(x2y + y3)

    2. Relevant equations

    3. The attempt at a solution

    The middle term is what's throwing me off. I can't put it in terms of y = vx (Dividing by x2 means that the y term gets screwed up. (It would turn it into yv2). Out of curiosity, I tried flat-out substituting y = vx (and turning dy/dx into x*dv/dx + v), but that didn't work either. It's obviously not exact, and I don't see how it could be made linear, so I'm not sure where to begin.
     
  2. jcsd
  3. Jul 21, 2014 #2
    Doesn't look like there's an elementary solution to this: http://www.wolframalpha.com/input/?i=dy/dx=x/(x^2y+y^3)
     
  4. Jul 21, 2014 #3
    The answer will involve the Lambert W function. Is this something you are familiar with? Basically, one arrives at something like ##z + \ln z = c## (or ##c = ze^z##) and needs to solve for ##z##.

    Start by substituting ##u = y^2## and then do another substitution. This should give you a separable equation which, after you solve it, gives you something like ##v - \ln v = g(x)##. This would be where the Lambert W function comes into play.

    Edit: If you don't like Lambert's W function, you can find a function ##h## such that ##x = h(y)##.
     
    Last edited: Jul 21, 2014
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