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Homework Help: ODE change of variable

  1. Sep 6, 2012 #1
    1. The problem statement, all variables and given/known data

    I have the ODE [itex] y' = f(\frac{y}{x}) [/itex], and I want to re-write this as a separable equation using the change of variable [itex] u = \frac{y}{x} [/itex]

    3. The attempt at a solution

    I use the chain rule to write [itex] y' = \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}
    = \frac{dy}{du}(-\frac{y}{x^2}) = -\frac{dy}{du}\frac{u^2}{y} = f(u) [/itex]

    which is a separable equation. However this seems to be wrong somehow because when I try using it to solve equations of the above form, I'm getting the wrong answer. Any help where I went wrong?
  2. jcsd
  3. Sep 6, 2012 #2
    If [itex]u=y/x[/itex] then [itex]y'=x\frac{du}{dx}+u[/itex]
  4. Sep 6, 2012 #3
    I don't see how I can use that to put the original equation [itex] \frac{dy}{dx} = f(\frac{y}{x}) [/itex] into separable form though. :\
  5. Sep 6, 2012 #4
    Won't you then have the equation:


    Ain't that separable?
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