# Homework Help: Polynomial differential operators

1. Apr 24, 2010

### sassie

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 $$\frac{1}{p(D)}e^{at}=\frac{1}{p^{(m)}(a)}t^me^{at}$$

where $$p^{(m)}(t)$$ is the $$m^{th}$$ derivative of p(t).

2. Relevant equations

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

Also, $$p(D)\frac{1}{p(D)}e^{at}=e^{at}$$

Then somehow I am meant to show that p(D)x(some value)=$$e^{at}$$ 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!