Find a linear differential operator that annihilates the given function. (use d for the differential operator.) e−x + 8xex − x2ex

 

Explanation:

Let 

.
= nth derivative of f

Note that 



So, (d + 1) annihilates . 

Note that 



Let 

For any real number  and positive integer n, 

      (1)

So, for g(x), . Thus, using equation (1),

     (2)

Since,  , 



Moreover, since 


     (3)

Hence based on equations (2) and (3),



Therefore, the linear operator  annihilates .