Question

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

the given function.

${\mathbf{x}}^{\mathbf{3}}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{5}\mathbf{x}\mathbf{)}$

Short answer

${\mathrm{D}}^5$

Step-by-step solution

Annihilator Operator

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

$\mathrm{L}\left(\mathrm{f}\right(\mathrm{x}\left)\right)=0$

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

Using the Differential operators

Differential operator that annihilates given polynomial is,

${\mathrm{x}}^3(1-5\mathrm{x})$

We can write as,

${\mathrm{x}}^3-5{\mathrm{x}}^4$

Since the given function contain polynomial of the highest power of 4 then differential ${\mathrm{D}}^5$will annihilate the given polynomial function,

${\mathrm{D}}^5({\mathrm{x}}^3-5{\mathrm{x}}^4)=0$

Hence the differential operator is${\mathrm{D}}^5.$