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Polynomial differential operators

  1. Apr 24, 2010 #1
    1. The problem statement, all variables and given/known data

    p(D) is a polynomial D operator of degree n>m. Suppose a is a m fold root of p(t)=0, but not a (m+1) fold root.

    Verify that [tex]\frac{1}{p(D)}e^{at}=\frac{1}{p^{(m)}(a)}t^me^{at}[/tex]

    where [tex]p^{(m)}(t)[/tex] is the [tex]m^{th}[/tex] derivative of p(t).

    2. Relevant equations

    For this question, we were told to use the exponential shift formula: [tex]\frac{1}{p(D)}e^{at}=e^{at}\frac{1}{p(D+a)}[/tex].

    Also, [tex]p(D)\frac{1}{p(D)}e^{at}=e^{at}[/tex]

    Then somehow I am meant to show that p(D)x(some value)=[tex]e^{at}[/tex] but I'm not sure of what to do from here.

    Also, because n>m, p(a) does not equal 0.

    3. The attempt at a solution

    As above.

    Your help is very much appreciated!
  2. jcsd
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