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Homework Help: Using the inverse hyperbolic tangent function to solve ODE

  1. Nov 16, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi all. I have to solve the differential equation [tex]\frac{dv}{dt} = g(1 - \frac{\rho}{g}v^2)[/tex].

    3. The attempt at a solution

    Apparently the solution should involve the inverse hyperbolic tangent function - with the equation in this form it should just be separable, correct? However, when separating variables I have to integrate the function [tex]\frac{dv}{1-\frac{\rho}{g}v^2}[/tex] which I am not sure how to go about. I think a substitution of some kind? Any tips would be appreciated.
  2. jcsd
  3. Nov 16, 2009 #2


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    Partial fractions: [itex]1-\frac{\rho}{g}v^2= (1- \sqrt{\frac{\rho}{g}}v)(1+ \sqrt{\frac{\rho}{g}}v)[/itex].
  4. Nov 16, 2009 #3


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    Or v=sqrt(g/rho)*tanh(u), if you want to stick with the hyperbolic function approach. You'll get the same answer, though it will look different.
  5. Nov 16, 2009 #4
    I see it now. Thanks guys!
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