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Polynomial annihilator

  1. Oct 27, 2009 #1
    1. The problem statement, all variables and given/known data
    is there a general method for obtaining polynomial annihilators?


    2. Relevant equations
    for stuff like: [tex] \beginarray {
    x $+ e^{x}sin(2x) $ \\
    7e^{x} $+ e^{x} $ \\
    e^{2x} $ + sin(5x) $ \\
    e^{3x} $ (3cos(2x) - $ sin(2x) )
    }
    [/tex]


    3. The attempt at a solution
    i know of how to do the second one (its a solution to the homogenous linear d.e. Whose auxilliary equation has the roots -1and 1, hence the auxilliary equation is factored from (r-1)(r+1)=0 and the corresponding annihilator; [itex] (D^2 - 1) [/itex] ) is there a general method ?
     
  2. jcsd
  3. Oct 27, 2009 #2

    lurflurf

    User Avatar
    Homework Helper

    There are a few. The brute force way is to take lots of derivatives (enough that all further derivatives will be linear combinations of previous ones). Then write the derivative operator as a matrix with the functions you want to annilate. The characteristic polynomial of this matix is an annihilator of the basic by cayley hamilton theorem. A slightly more efficient method is to take each function and write it in the form
    p(x)exp(c x)cos(ax+b) where p(x) is a polynomial and a, b, and c are constants
    then (D-(c+ai))^(n+1) where n is the degree of p will be a factor of the annihilator
    so take the least common multiple of all such factors and if a real annihilator is wanted also include complex conjugates that is use
    (D-(c+ai))^(n+1)(D-(c-ai))^(n+1) instead of(D-(c+ai))^(n+1) whenever a is not zero
     
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