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Math Help - Question with Diff Eq

  1. #1
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    Question with Diff Eq

    I haven't had Diff Eq, and in my notes for an unrelated class, part of a way to solve a solution was with the following diff eq.

    (d^2)x/d(t^2) + a^2 * x = 0 | a=constant; t=0; dx/dt = 0


    I really don't know what to do.

    Any help is greatly appreciated.

    Thanks
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Katzenjammer View Post
    I haven't had Diff Eq, and in my notes for an unrelated class, part of a way to solve a solution was with the following diff eq.

    (d^2)x/d(t^2) + a^2 * x = 0 | a=constant; t=0; dx/dt = 0


    I really don't know what to do.

    Any help is greatly appreciated.

    Thanks
    Supposing a solution of the form x=e^{rt}, we see that our auxiliary equation becomes r^2+a=0.

    However, I ask this question: is a a positive or negative constant?

    I don't want to continue on until this is clarified.
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    Supposing a solution of the form x=e^{rt}, we see that our auxiliary equation becomes r^2+a=0.

    However, I ask this question: is a a positive or negative constant?

    I don't want to continue on until this is clarified.
    I have no idea, it was at the end of class and my Dynamics teacher said "finish this for next class."

    I'm assuming it's a positive, since it's acceleration.
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Katzenjammer View Post
    I have no idea, it was at the end of class and my Dynamics teacher said "finish this for next class."

    I'm assuming it's a positive, since it's acceleration.
    Quote Originally Posted by Chris L T521 View Post
    Supposing a solution of the form x=e^{rt}, we see that our auxiliary equation becomes r^2+a=0.

    However, I ask this question: is a a positive or negative constant?

    I don't want to continue on until this is clarified.
    Ok, continuing on...

    The solution to the characteristic equation is r^2=-a\implies r=\pm i\sqrt{a}.

    So it follows that the general solution to the equation is x(t)=c_1\cos\!\left(\sqrt{a}t\right)+c_2\sin\!\lef  t(\sqrt{a}t\right)

    Are you sure there isn't another initial condition? Since its a second order equation, there should be one more initial condition.
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  5. #5
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    Just peeked back my notes, he only gave the three initial conditions.

    It wouldn't surprise me if he left something out, he's known to do that. :-p

    Thanks for your help, though.
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