Question

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

the given function.

$\mathit{x}\mathbf{+}\mathbf{3}\mathit{x}{\mathit{e}}^{\mathbf{6}\mathbf{x}}$

Short answer

$D^2{(D-6)}^2$

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,

$x+3x{e}^{6x}$

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 $x^n{e}^{\alpha \chi }$is annihilated by ${\left(D-\alpha \right)}^{n+1}$, then we annihilates the components of the given function as,

x¹ is annihilated by the linear differential operator D².

3xe^6x is annihilated by the linear differential operator (D - 6)², where $\alpha =6$and n = 1.

Then we can annihilates the given function as

$D^2{\left(D-6\right)}^2\left(x+3x{e}^{6x}\right)=0$

Hence, the used differential operator is${\mathit{D}}^{\mathbf{2}}{\left(D-6\right)}^{\mathbf{2}}$