# Polynomial annihilator

1. Oct 27, 2009

### Mechdude

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: $$\beginarray { x + e^{x}sin(2x)  \\ 7e^{x} + e^{x}  \\ e^{2x}  + sin(5x)  \\ e^{3x}  (3cos(2x) -  sin(2x) ) }$$

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; $(D^2 - 1)$ ) is there a general method ?

2. Oct 27, 2009

### lurflurf

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