Is There a General Method for Finding Polynomial Annihilators?

[Mechdude]

Homework Statement

is there a general method for obtaining polynomial annihilators?

Homework Equations

for stuff like: $$\beginarray {<br /> x + e^{x}sin(2x) \\<br /> 7e^{x} + e^{x} \\ <br /> e^{2x} + sin(5x) \\<br /> e^{3x} (3cos(2x) - sin(2x) ) <br /> }$$

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 ?

 

[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