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Simple problem in Mechanics, weird differential equation

  1. Jan 27, 2008 #1
    There is a problem I couldn't figure out , it says :

    it says that a particle of mass m moves along a straight line and is acted on by a retarding force (one always directed against the motion) F=b*exp(a*v(t)),
    b, a are constants and v is the velocity.

    At t=0 it is moving with velocity V

    and I am aked to solve the differential equation that results from this to get a function of v(t).

    I found that the differential equation that has to be solved is :

    dv/dt = (b/m)*exp[a*v]

    so this is like solving a non-linear differential equation of the form y'=exp(y)

    How do you do that??
  2. jcsd
  3. Jan 27, 2008 #2


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    Do you know how to separate variables ?
  4. Jan 27, 2008 #3


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    You have simply to re-write the equation in terms of differentials and you will be able to integrate:

    [tex] \frac{du}{dt} = f(u)[/tex]
    [tex] du = f(u) dt[/tex]
    and then
    [tex] \frac{du}{f(u)} = dt[/tex]
    you can then integrate:
    [tex] \int \frac{du}{f(u)} = t + C[/tex]
    which gives you [tex]u(t)[/tex] implicitly.

    In your case you should get [tex]v(t)[/tex] in terms of a logarithm of t.
  5. Jan 27, 2008 #4

    I understood everything. Thank you so much
  6. Jan 27, 2008 #5
    Just a note! :smile:

    Since you have a retarding force, the ODE should be

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