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ODEs- annihilator method

  1. Mar 12, 2010 #1
    1. The problem statement, all variables and given/known data
    How can I annihilate the following ?
    4e-2t*cos(2t)

    2. Relevant equations



    3. The attempt at a solution

    I know that if I want to annihilate e-t
    it would be (D-1) and to annihilate cos(2t) it would be (D2+22)

    but what happens if they are multiplied ?
    how do I annihilate this ?
    I tried something and I'm not sure it's correct but I got
    (D2+4)/(D-2)
    ?
     
  2. jcsd
  3. Mar 13, 2010 #2

    Mark44

    Staff: Mentor

    If you wanted to annihilate, for example, etsin(3t) or etcos(3t), the annihilator would be D2 - 2D + 10.

    The characteristic equation for the differential equation y'' - 2y' + 10y = 0, or (D2 - 2D + 10)y = 0, is r2 - 2r + 10 = 0. The roots of this latter equation are r = 1 +/- 3i. The connection between etsin(3t) and etcos(3t) on the one hand, and 1 +/- 3i on the other, is not coincidental.
     
  4. Mar 17, 2010 #3
    I am having a little problem following your explanation.

    Say this is your differential equation:
    y'' - 2y' + 10y =etcos(3t)

    or for the left part only
    (D2 - 2D + 10)y
    and what you are basically saying is that the left part of the equation => (D2 - 2D + 10) annihilates etcos(3t)

    so it should look this way :
    (D2 - 2D + 10)*(D2 - 2D + 10)y =(D2 - 2D + 10)*etcos(3t)

    <=>(D2 - 2D + 10)2y=0

    Thank you.
     
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