Question

In Problems 15–26 find a linear differential operator that annihilates

the given function.

$\mathbf{13}\mathit{x}\mathbf{+}\mathbf{9}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathit{s}\mathit{i}\mathit{n}\mathbf{4}\mathit{x}$

Short answer

$D^3\left(D^2+16\right).$

Step-by-step solution

Annihilator Operator

If L is a linear differential operator with constant coefficients and f is a sufficiently differentiable function such that

$L\left(f\right(x\left)\right)=0$

then L is said to be an annihilator of the function.

Using the Differential operators

We have the function,

$13x+9x^2-\sin 4x$

And we have to annihilate it using a linear differential operator as the following technique:

Xⁿ is being annihilated by the linear differential operator Dⁿ⁺¹ and $\sin \beta \chi$is being annihilated by the linear differential operator $\left(D^2+{\beta }^2\right),$then we annihilates the components of the given function as,

$13x+9x^2$is annihilated by the linear differential operator D³, where it is a polynomial function with a higher power n = 2.

$-\sin 4x$is annihilated by the linear differential operator $\left(D^2+4^2\right)=\left(D^2+16\right)$, where $\beta =4.$

Then we can annihilates the given function as

$D^3\left(D^2+16\right)\left(13x+9x^2-\sin 4x\right)=0$

Hence, the used differential operator is ${\mathit{D}}^{\mathbf{3}}\left(D^2+16\right)$.