What is the annihilator operator of the function $ \ f(x)= x(\sin x+3x-x^{-1})-\cos x \ $

Question

What is the annihilator operator of the function $ \ f(x)= x(\sin x+3x-x^{-1})-\cos x \ $ , where the operator is $ \ \frac{d}{dt}\equiv D$ . $$ $$ My approach is - $ \ f(x)=x \sin x+3x^{2}-1-\cos x \ $ . \begin{align} (i) \ annihilator \ of \ \ 1 \ \ is \ D, \\ (ii) \ annihilator \ of \ \cos x \ \ is \ \ (D^{2}+1), \\ (iii) \ annihilator \ of \ 3x^{2} \ is \ D^{3}, \\ (iv) \ annihilator \ of \ x \sin x \ is \ what \ ? \end{align}. Please help me finding the annihilator of $ \ x \sin x \ $. Because after that we have to take make product of all annihilator to find the annihilator of the main equation. I can do that . But please help me finding the annihilator of $ \ x \sin x \ $ . Any help is appreciated.

Answer 1

Hint:

Observe that $$(D^2+1)(x\sin x)=2\cos x-x\sin x+x\sin x=2\cos x$$ So, think about what operator anihilates last function.