91Ó°ÊÓ

Consider the given function:

$f\left(x\right)=x^2-2x+5$

Since f(x) is a polynomial function of second order then it is annihilated by:

$D^{2+1}=D^3$

Consider the given function:

$f\left(x\right)=e^{3x}+x-1.$

Let $f_1\left(x\right)=e^{3x}$and $f_2\left(x\right)=x-1$. Observe that D - 3 annihilates f₁(x) and f₂(x) is annihilated by D².

Hence, the composite operator $D^2(D-3)$annihilates both f₁(x) and f₂(x) so it annihilates

f₁(x) + f₂(x).

Consider the given function:

$f\left(x\right)=x\sin 2x$

This function is annihilated by:

${\left(D^2+2^2\right)}^2={\left(D^2+4\right)}^2.$

Consider the given function:

$f\left(x\right)=x^2{e}^{-2x}\cos 3x$

This function is annihilated by:

[(D + 2)² + 3² ]³ = (D² + 4D + 4 + 9)³= (D² + 4D + 13)³

Consider the given function:

$f\left(x\right)=x^2-2x+x{e}^{-x}+\sin 2x-\cos 3x$

Let $f_1\left(x\right)=x^2-2x,f_2\left(x\right)=x{e}^{-x},f_3\left(x\right)=\sin 2x$and $f_4\left(x\right)=\cos 3x$.

Observe that D³ annihilates f₁(x) and f₂(x) is annihilated by (D + 1)².

Furthermore, we see that D²+ 2² annihilates f₃(x) and f₄(x) is annihilated by D²+ 3² . Hence, the composite operator D³(D + 1)² (D² + 2²) (D² + 3²) =D³(D + 1)² (D² + 4) (D² + 9)

annihilates f₁(x), f₂(x), f₃(x) and f₄(x) so it annihilatesf₁(x) + f₂(x) + f₃(x) - f₄(x)