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Differential Equation - Determining frequency of beats/rapid oscillations

  1. May 11, 2009 #1
    1. The problem statement, all variables and given/known data

    y'' + 6y = 2cos3t

    a)Determine frequency of the beats
    b)Determine frequency of the rapid oscillations
    c)Use the information from parts a) and b) to give a rough sketch of a typical solution

    2. Relevant equations



    3. The attempt at a solution

    Not sure how to do the first 2 parts. I know the natural period is [tex]\frac{2 \pi}{B}[/tex] and natural frequency is [tex]\frac{B}{2 \pi}[/tex]
     
  2. jcsd
  3. May 12, 2009 #2

    Mark44

    Staff: Mentor

    The characteristic equation of the homogeneous equation is r2 + 6 = 0, so its solutions are r = +/- i[itex]\sqrt{6}[/itex]. This means that
    [tex]y_h = Ae^{i\sqrt{6}t} + Be^{-i\sqrt{6}t}[/tex]
    It's more convenient to write a different linear combination of these to get
    [tex]y_h = c_1cos(\sqrt{6}t) + c_2sin(\sqrt{6}t)[/tex]

    For the nonhomogeneous equation, look for a solution yp = Acos(3t) + Bsin(3t).

    The general solution will be yh + yp.

    For a, you'll have to figure out how the functions in the homogeneous solution interact with what I'm pretty sure will be the one particular solution function, and how often the waves of the two periodic parts reinforce each other (the beats).

    For b, you need to find the rapid oscillation frequency. You have in essence a long wave (low frequency) imposed on a short wave (high frequency).

    Hope that helps. It's getting late where I am, so I'm going to turn in.
     
  4. May 12, 2009 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Since there is no "B" in the statement of your problem, that makes no sense! Mark44 has already pointed out that the general solution to the associated homogeneous equation is [itex]A cos(\sqrt{6}t)+ B sin(\sqrt{6}t)[/itex]. That tells you that the natural period is [itex]2\pi/\sqrt{6}[/itex].

    Trig identity: cos(s+ t)= cos(s)cos(t)- sin(s)sin(t) so if you have a formula of the form Acos(2\pi t/\sqrt{6})+ Bcos(3t) you can write that, by careful manipulation of A and B, as [itex]cos((2\pi/\sqrt{6}+ 3)t)[/itex].
     
  5. May 12, 2009 #4
    It's not a B. It's supposed to be the greek uppercase Beta.

    as in - [itex]A cos(Bt)+ B sin(Bt)[/itex]. It's how my textbook/prof write it.

    In this case B (Beta) = [itex]\sqrt{6}[/itex]
     
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