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Math Help - Ordinary Differential equation help needed.

  1. #1
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    Ordinary Differential equation help needed.

    w = omega, just cant get it on.

    ii) d2i/dt2 + 900i = A0 sin (wt)

    Assuming that w2 not equal to 900, determine the current i in terms of the parameters (w and A0) and the variable t when the initial conditions are

    i (0) = di/dt (0) = 0.

    iii) Repeat part (ii) for w2 = 900

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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by naisy18 View Post
    w = omega, just cant get it on.

    ii) d2i/dt2 + 900i = A0 sin (wt)

    Assuming that w2 not equal to 900, determine the current i in terms of the parameters (w and A0) and the variable t when the initial conditions are

    i (0) = di/dt (0) = 0.

    iii) Repeat part (ii) for w2 = 900

    Help required people.
    i was able to wake up early..

    \frac{d^2i}{dt^2} + 900i = A_0 sin (\omega t)

    let us try solving the general case: \frac{d^2y}{dt^2} + \omega ^2y = 0

    consider the operator D = \frac{d}{dt}, then we can transform the equation into D^2 y + \omega ^2 y = (D^2 + \omega ^2)y = (D - \omega i)(D + \omega i)y = 0, where i^2 = -1 (i have to change our variable to y because we need the imaginary number i).

    so, (D - \omega i)y = \left( {\frac{d}{dt} - \omega i} \right)y=0

    but the middle one is also equal to e^{i\omega t} \frac{d}{dt}\left( {e^{-i\omega t}} \right) \implies y_1 = e^{i\omega t} = cos (\omega t) + i sin (\omega t)

    also using similar arguments, it can be shown that (D + \omega i)y \implies y_2 = cos (\omega t) - i sin (\omega t)

    if we take \Upsilon _1 = \frac{y_1 + y_2}{2} = cos (\omega t) and \Upsilon _2 = \frac{y_1 - y_2}{2i} = sin (\omega t), then we have the general solution:

    y(t) = c_1 cos (\omega t) + c_2 sin (\omega t) ---> 1

    now, to solve for the value of c_1 and c_2, just consider that y(0) = 0 = \frac{dy}{dt}(0)

    equate equation 1 to 0.
    solve for the first derivative of that equation 1 and equate to 0..
    now you have two equations with two unknowns .. can you contnue it?
    i'm late for my class...
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