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Homework Help: Differential Equations - Simple Harmonic Oscillation

  1. Jun 28, 2007 #1
    1. The problem statement, all variables and given/known data

    Consider y''(t)+(k/m)*y = 0 for simple harmonic oscillation

    A) Under what conditions on Beta is y(t)=cos(Beta*t) a solution?

    B) What is the period of this solution?

    C) Sketch the solution curve in the yv-plane associated with this solution (Hint: y^2 + (v/Beta)^2)

    For A, I had:

    dy/dt = v

    dv/dt = -(k/m)*y

    Found y' and y''

    y'(t) = -Beta*sin(Beta*t) y''(t)=-Beta^2*cos(Beta*t)

    Plugged those into the given equation and found Beta = +/- sqrt(k/m)

    y(t) = cos (+/-sqrt(k/m)*t) -> answers for A

    For B, I found the period to be T = 2*pi / Beta

    For C, I found that y^2 + (v/Beta)^2 = cos^2(Beta*t) + Beta^2*sin^2(Beta*t)/Beta^2

    My problem now is drawing this. I think, from remembering equations of shapes, that this is an ellipse, stretching in the v direction.

    I am not sure, but I think the max and min points of the ellipse are:
    y = 0, v = +/- sqrt(k/m)
    v = 0, y = +/- 1

    I also think that the direction of the field is counter-clockwise.

    I don't know if part B and C were done totally right and am a bit confused about finding the direction of the phase field/solution.

  2. jcsd
  3. Jun 28, 2007 #2


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    Science Advisor

    Pretty good- since cosine is an evern function, you don't need the "+/-", both signs give the same function, but, yes, Beta= sqrt(k/m).

    Wouldn't it be better to replace Beta by the value you found in (a)?

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