Question

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

the given function.

$\mathbf{8}\mathit{x}\mathbf{-}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{10}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{5}\mathbf{}\mathit{x}\mathbf{.}$

Short answer

$D^2\left(D^2+1\right)\left(D^2+25\right)$

Step-by-step solution

Annihilator Operator

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

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

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

Using the Differential operators

We have the function,

$8x-\sin x+10\cos 5x$

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

${\mathrm{x}}^{\mathrm{n}}$is being annihilated by the linear differential operator $D^{n+1}$and $\mathrm{cos\beta x}$or $\mathrm{sin\beta x}$is being annihilated by the linear differential operator $(D^2+{\beta }^2)$, then we annihilates the components of the given function as,

$8{\mathrm{x}}^1$is annihilated by the linear differential operator ${\mathrm{D}}^2$, where n = 1.

$-\sin \mathrm{x}$is annihilated by the linear differential operator $({\mathrm{D}}^2+1)$, where $\mathrm{\beta }=1$.$10\cos 5\mathrm{x}$is

annihilated by the linear differential operator $(D^2+25)$, where$\beta =5$.

Then we can annihilates the given function as

${\mathrm{D}}^2({\mathrm{D}}^2+1)({\mathrm{D}}^2+25)(8\mathrm{x}-\sin \mathrm{x}+10\cos 5\mathrm{x})=0$

Hence, the used differential operator is${\mathit{D}}^{\mathbf{2}}\mathbf{(}{\mathit{D}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}{\mathit{D}}^{\mathbf{2}}\mathbf{+}\mathbf{25}\mathbf{)}$