Question

In Problems 11–14 verify that the given differential operator annihilates the indicated functions.

$\mathbf{2}\mathbf{D}\mathbf{-}\mathbf{1}\mathbf{;}\mathbf{y}\mathbf{=}\mathbf{4}{\mathbf{e}}^{\mathbf{x}\mathbf{/}\mathbf{}\mathbf{2}}$

Short answer

Proved

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.

Verify that the given differential operator annihilated the indicated function.

$2\mathrm{D}-1:\mathrm{y}=4{\mathrm{e}}^{\frac{\mathrm{x}}{2}}$

We proof that L(y)=0 where L is annihilates of the function.

We know that $\alpha =\frac{1}{2}$and n = 1 and so it follows that,

$$\begin{gathered} (\mathrm{D}-\frac{1}{2})4{\mathrm{e}}^{\frac{\mathrm{x}}{2}}=0 \\ (2\mathrm{D}-1)4{\mathrm{e}}^{\frac{\mathrm{x}}{2}}=0 \\ 2\mathrm{D}\left(4{\mathrm{e}}^{\frac{\mathrm{x}}{2}}\right)-4{\mathrm{e}}^{\frac{\mathrm{x}}{2}}=0 \\ 4{\mathrm{e}}^{\frac{\mathrm{x}}{2}}-4{\mathrm{e}}^{\frac{\mathrm{x}}{2}}=0 \\ 0=0 \end{gathered}$$

So, RHS=LHS.

Hence the differential operator annihilates the indicated function.