Question

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

$\mathbf{(}\mathbf{D}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{(}\mathbf{D}\mathbf{+}\mathbf{5}\mathbf{)}\mathbf{:}\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\mathbf{+}\mathbf{3}{\mathbf{e}}^{\mathbf{-}\mathbf{5}\mathbf{x}}$

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

L(f(x))=0

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

Using the Differential operators

To verify that the differential operator $(\mathrm{D}-2)(\mathrm{D}+5)$annihilates the function $\mathrm{y}={\mathrm{e}}^{2\mathrm{x}}+3{\mathrm{e}}^{-5\mathrm{x}}$, we have to verify that $(\mathrm{D}-2)(\mathrm{D}+5)\mathrm{y}=0$as the form L(y)=0 by applying the differential operator to the both sides of function and differentiate as the following:

localid="1667898444782" $$\begin{gathered} \mathit{(}D\mathit{-}\mathit{2}\mathit{)}\mathit{(}D\mathit{+}\mathit{5}\mathit{)}y\mathit{=}\mathit{(}D\mathit{-}\mathit{2}\mathit{)}\mathit{(}D\mathit{+}\mathit{5}\mathit{)}\mathit{(}{e}^{\mathit{2}x}\mathit{+}\mathit{3}{e}^{\mathit{-}\mathit{5}x}\mathit{)} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{=}\mathit{(}{D}^{\mathit{2}}\mathit{+}\mathit{5}D\mathit{-}\mathit{2}D\mathit{-}\mathit{10}\mathit{)}\mathit{(}{e}^{\mathit{2}x}\mathit{+}\mathit{3}{e}^{\mathit{-}\mathit{5}x}\mathit{)} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{=}\mathit{(}{D}^{\mathit{2}}\mathit{+}\mathit{3}D\mathit{-}\mathit{10}\mathit{)}\mathit{(}{e}^{\mathit{2}x}\mathit{+}\mathit{3}{e}^{\mathit{-}\mathit{5}x}\mathit{)} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{=}{D}^{\mathit{2}}\mathit{\left(}{e}^{\mathit{2}x}\mathit{\right)}\mathit{+}\mathit{3}{D}^{\mathit{2}}{e}^{\mathit{-}\mathit{5}x}\mathit{+}\mathit{3}D{e}^{\mathit{2}x}\mathit{+}\mathit{9}D{e}^{\mathit{-}\mathit{5}x}\mathit{-}\mathit{10}{e}^{\mathit{2}x}\mathit{-}\mathit{30}{e}^{\mathit{-}\mathit{5}x} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{=}\mathit{2}D{e}^{\mathit{2}x}\mathit{-}\mathit{15}D{e}^{\mathit{-}\mathit{5}x}\mathit{+}\mathit{6}{e}^{\mathit{2}x}\mathit{-}\mathit{45}{e}^{\mathit{-}\mathit{5}x}\mathit{-}\mathit{10}{e}^{\mathit{2}x}\mathit{-}\mathit{30}{e}^{\mathit{-}\mathit{5}x} \\ \mathit{=}\mathit{4}{e}^{\mathit{2}x}\mathit{+}\mathit{75}{e}^{\mathit{-}\mathit{5}x}\mathit{+}\mathit{6}{e}^{\mathit{2}x}\mathit{-}\mathit{45}{e}^{\mathit{-}\mathit{5}x}\mathit{-}\mathit{10}{e}^{\mathit{2}x}\mathit{-}\mathit{30}{e}^{\mathit{-}\mathit{5}x} \\ \mathit{=}\mathit{(}\mathit{4}\mathit{+}\mathit{6}\mathit{-}\mathit{10}\mathit{)}{e}^{\mathit{2}x}\mathit{+}\mathit{(}\mathit{75}\mathit{-}\mathit{45}\mathit{-}\mathit{30}\mathit{)}{e}^{\mathit{-}\mathit{5}x} \\ =0+0 \\ =0 \end{gathered}$$

Hence the differential operator annihilates the indicated function.