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Math Help - Help with Undetermined Coefficients problem

  1. #1
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    Help with Undetermined Coefficients problem

    I need help with the following problem:

    y'' + 4y = cos^2(t)

    Here is what I have done so far
    y'' + 4y = (1/2) + (1/2)cos(2t) (power reduction formula)
    Y(t) = A + B*sin(2t) + C*cos(2t)
    Y'(t) = 2B*cos(2t) - 2C*sin(2t)
    Y''(t) = -4B*sin(2t) - 4C*cos(2t)

    Pluging in Y(t) and Y''(T) into the original equation, I get:
    [-4B*sin(2t) - 4C*cos(2t)] + 4[A + B*sin(2t) + C*cos(2t)] = (1/2) + (1/2)cos(2t)
    4A = (1/2) + (1/2)cos(2t)

    As you can see, the B and C terms cancel each other out, leaving only an A term. What should I do from here? Is there anything I did wrong?
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  2. #2
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    The first thing you need to do is solve the homogeneous DE y'' + 4y = 0. Once you have done that you should see a problem when you try to solve for the non-homogeneous y'' + 4y = cos^2(t).
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  3. #3
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    *** Lecture Warning ***

    Posting the same problem in multiple forums is not in good taste. We understand your need for urgency, but you need to understand the need not to waste the time of volunteers. Since I already answered this sufficiently on another forum, what might we say of any time spent answering on this forum? I don't wish to carry this too far, as frequently it is the case that multiple and varied responses generate interesting and useful discussion. Keep in mind that the time of volunteers from whom you are requesting time is a finite and valuable resource. We do NOT want to waste it.

    *** End of Lecture **

    Please include D*t*sin(2t) and E*t*cos(2t) in your definition of Y(t) and you'll get it. If too much cancels out, this is always a good next step. Complicate the original definition.
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