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I Second order DE with Sine function

  1. Apr 7, 2017 #1
    I have this second order differential equation but I'm stumped as to how to solve this since the zeroth order term has a Sine function in it and the variable is embedded.

    ##\ddot y(t) + 3H (1+Q) \dot y(t) -m^2 f \sin(\frac{y(t)}{f}) = 0##

    ##H~##, ##~Q~##, ##~m~##, and ##~f~## are just constants.

    I even tried to use DSolve in mathematica but there is an error. How do I solve this? Can anyone guide me with this problem?
    Last edited: Apr 7, 2017
  2. jcsd
  3. Apr 7, 2017 #2


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    Google damped pendulum differential equation
    [edit] and there's always this
  4. Apr 8, 2017 #3
    Are you pertaining to the small angle approximation? I know that is a possibility but the problem is that ##\theta = \frac{y[t]}{f} \approx 1##.
  5. Apr 8, 2017 #4


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    Small angle is the first approach. The article in the link discusses the full equation in section 3.
  6. Apr 8, 2017 #5
    If you're interested in non-linear equations, I recommend this book which includes a step-by step solution of the non-linear pendulum:

  7. Apr 9, 2017 #6
    I still can't find the solution to my given equation, can you kindly give me a hint/note onto where I should look?
  8. Apr 9, 2017 #7
    I was referring to the equation ## y''=k\sin(y)##. That's not yours but if you first study how this one is solved exactly in terms of elliptic functions, then perhaps you can adapt the method to yours. It's been a while that I've studied it (using the book I quoted) and I no longer have the book and don't recall exactly how it's done.
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