If I have a linear, non-homogeneous differential equation with a function like on the right-side, one of the standard methods is to use an annihilater to transform it to a homogeneous equation.

The operator annihilates functions of this form. However, it's also true that annihilates fucntions of the form . Using and and gives the annihilater , so the corresponding auxiliary equation will contain 2 repeated roots for these factors. However, the annihilater implies one root.

I guess in general you can annihilate functions many ways, but using different annihilaters will give different auxiliary equations right?

EDIT: Wait, is an assumption when deriving the operator, so what I'm doing doesn't make any sense. It also applies to the case that the function is a sine instead of a cosine, which would require a different value of too. However, The differential operator I obtained still seems to work. I tested it.