Question

find a differential operator that annihilates the given function.

$x^2{e}^{-x}sin2x$

Short answer

${\left(D^2+2D+5\right)}^3$is the differential operator that annihilates the given function.

Step-by-step solution

Any nonhomogeneous term of the form f(x)=xkeαxcosβx OR xkeαxsinβx satisfies (D-α)2+β2m[f]=0for K=0,1,...,m-1

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

Hence${\left[{(D+1)}^2+2^2\right]}^3\left[f\right]=0$

Then${\left[{(D+1)}^2+2^2\right]}^3={\left(D^2+2D+1+4\right)}^3={\left(D^2+2D+5\right)}^3$is the differential operator that annihilates the given function.