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Math Help - Undetermined coefficients

  1. #1
    Junior Member
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    Undetermined coefficients

    \frac{d^2x}{dt^2} + \omega^2x = F_0sin(\omega t), x(0) = 0 , x'(0) = 0

    Because this problem is not in term of x and y. Then I assume x is t and y is x.

    For general solution,
    x = x_c + x_p

    m^2 + \omega^2 = 0

    m = i\sqrt{\omega}

    x_c = e^t(c_1sin(\sqrt{\omega}t) + c_2cos(\sqrt{\omega}t))

    And for particular solution,
    x_p = Asin(\omega t)

    x'_p = Acos(\omega t)

    x''_p = -Asin(\omega t)

    Substitute into \frac{d^2x}{dt^2} + \omega^2x form,

    -Asin(\omega t) + \omega^2Asin(\omega t) = F_0sin(\omega t)

    Then, A = \frac{F_0}{\omega^2 - 1}

    Finally, x = e^t(c_1sin(\sqrt{\omega}t) + c_2cos(\sqrt{\omega}t)) + \frac{F_0}{\omega^2 - 1}sin(\omega t)

    My question is how x(0) = 0 and x'(0) = 0 involve with this problem?
    Thank you.

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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by noppawit View Post
    \frac{d^2x}{dt^2} + \omega^2x = F_0sin(\omega t), x(0) = 0 , x'(0) = 0

    Because this problem is not in term of x and y. Then I assume x is t and y is x.

    For general solution,
    x = x_c + x_p

    m^2 + \omega^2 = 0

    m = i\sqrt{\omega}

    x_c = e^t(c_1sin(\sqrt{\omega}t) + c_2cos(\sqrt{\omega}t))

    And for particular solution,
    x_p = Asin(\omega t)

    x'_p = Acos(\omega t)

    x''_p = -Asin(\omega t)

    Substitute into \frac{d^2x}{dt^2} + \omega^2x form,

    -Asin(\omega t) + \omega^2Asin(\omega t) = F_0sin(\omega t)

    Then, A = \frac{F_0}{\omega^2 - 1}

    Finally, x = e^t(c_1sin(\sqrt{\omega}t) + c_2cos(\sqrt{\omega}t)) + \frac{F_0}{\omega^2 - 1}sin(\omega t)

    My question is how x(0) = 0 and x'(0) = 0 involve with this problem?
    Thank you.

    They are the initial conditions and determine the values of the arbitrary constants c_1 and c_2 (that is if the general solution was correct which it is not, the solution to the homogeneous equation is wrong ( m=\pm i \omega) as is the particular integral)

    Spoiler:

    x={\it k_1}\,\sin \left(t\,w\right)+{\it k_2}\,\cos \left(t\,w\right)-{{F_0\,t\,\cos \left(t\,w \right)}\over{2\,w}}

    CB
    Last edited by CaptainBlack; July 10th 2009 at 03:08 AM.
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