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Math Help - ODE Problem

  1. #1
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    ODE Problem

    Is anyone able to do this?
    Im having lots of problems
    Attached Thumbnails Attached Thumbnails ODE Problem-ode.png  
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  2. #2
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    Re: ODE Problem

    For \displaystyle \frac{d^2i}{dt^2}+400i= 0 what is the charactersitic equation? You will need to solve this equation to contiune.

    Hint: Its a quadratic
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  3. #3
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    Re: ODE Problem

    There's probably an easier way to do this, but one valid and straightforward method is to convert to the system

    \left[\begin{array}{c}x_1\\x_2\end{array}\right]'=\left[\begin{array}{cc}0&1\\-400&0\end{array}\right]\left[\begin{array}{c}x_1\\x_2\end{array}\right],

    where i(t)=x_1. We use this to find the fundamental matrix

    X(t)=\left[\begin{array}{cc}-\cos 20t&(\sin 20t)/20\\20i\cos 20t&i\sin 20t\end{array}\right]

    and determine i(t)=a(-\cos 20t)+b(\sin 20t)/20 for some constants a,b\in\mathbb{R}. It's easy to verify that the solution to the IVP in (i) is then given by i(t)=5\cos 20t.
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    Re: ODE Problem

    Hi. Thanks for the reply
    I should have been more specific sorry. its part ii) that Im really stuck with. sorry
    for part i) I put the values into the quadratic formula and then with the obtained values substituted them into general solution giving me the desired graph that i was looking for
    If you could help me with part ii) that would be great.
    thanks
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  5. #5
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    Re: ODE Problem

    Quote Originally Posted by Bullet View Post
    Hi. Thanks for the reply
    I should have been more specific sorry. its part ii) that Im really stuck with. sorry
    for part i) I put the values into the quadratic formula and then with the obtained values substituted them into general solution giving me the desired graph that i was looking for
    If you could help me with part ii) that would be great.
    thanks
    For part ii) you have solved the homogeneous DE. Now solve the nonhomogeneous using variation of parameters. The general solution is the sum of the homogeneous and nonhomogeneous solutions.
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