(adsbygoogle = window.adsbygoogle || []).push({}); 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!

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# Polynomial differential operators

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