Question

find a differential operator that annihilates the given function

$x^2{e}^x-xsin4x+x^3$

Short answer

$D^4(D-1{)}^3{\left(D^2+16\right)}^2$is the differential operator that annihilates the given function.

Step-by-step solution

Any nonhomogeneous term of the form f(x)=xkeαxcosβx OR role="math" localid="1663946799201" xkeαxsinβxsatisfiesrole="math" localid="1663946761280" (D-α)2+β2m[f]=0for K=0,1,2,...,m-1

Let the function be$f\left(x\right)=x^2{e}^x-x\sin 4x+x^3$

Let$g\left(x\right)=x^2{e}^x$

Then

$(D-1{)}^3\left[g\right]=0$

Let$h\left(x\right)=x{e}^{-5x}\sin 3x$

Then

${\left(D^2+4^2\right)}^2\left[h\right]=0$

Let$i\left(x\right)=x^3$

Then

$D^4\left[i\right]=0$

Hence

$$\begin{gathered} (D-1{)}^3{\left(D^2+16\right)}^2{D}^4\left[f\right]=0 \\ \Rightarrow D^4(D-1{)}^3{\left(D^2+16\right)}^2\left[f\right]=0 \end{gathered}$$

Then$D^4(D-1{)}^3{\left(D^2+16\right)}^2$ is the differential operator that annihilates the given function.