8.14.4. Comment

The annihilator method works whenever we can find a constant coefficient opeator A to annihilate the nonhomogeneous term R.  For convenience, we list below the annihilators for some commonly encountered functions.

 

Function

Annihilator

x m1

D m

e αx

Dα

x m1 e αx

( Dα ) m

cosβx  or sinβx

D 2 + β 2

x m1 cosβx  or x m1 sinβx

( D 2 + β 2 ) m

e αx cosβx  or e αx sinβx

D 2 2αD+ α 2 + β 2

x m1 e αx cosβx  or x m1 e αx sinβx

( D 2 2αD+ α 2 + β 2 ) m