Find an annihilator for each function:

1. f(x)=x⁴-7x²+2
2. f(x)=e^2x
3. f(x)=4e^2x-xe^2x+x²e^2x
4. (harder) f(x)=9xe^2xcos(3x)-5xe^2xsin(3x) (hint: look at 5 above)
5. f(x)=Acos(kx)+Bsin(kx)

Solution:

1. f(x)=x⁴-7x²+2 is annihilated by D⁵ (or higher any power)
2. f(x)=e^2x is annihilated by D-2 (or any higher power of D-2)
3. f(x)=4e^2x-xe^2x+x²e^2x is annihilated by (D-2)³ (or higher any power of D-2 greater than 3)
4. f(x)=9xe^2xcos(3x)-5xe^2xsin(3x) the A is 2, and the B is 3 and n-1=1, so n=2, so the annihilator is (D²-2AD+(A²+B²))²=(D²-4D+13)²
5. f(x)=Acos(kx)+Bsin(kx) is annihilated by (D²+k²)

I would encourage you to check one or two of these to see how the annihilator works.