Question

find a differential operator that annihilates the given function.

$x{e}^{3x}cos5x$

Short answer

${\left(D^2-6D+34\right)}^2$is the differential operator that annihilates the given function.

Step-by-step solution

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

Let the function be$f\left(x\right)=x{e}^{3x}\cos 5x$

Hence${\left[{(D-3)}^2+5^2\right]}^2\left[f\right]=0$

Then${\left[{(D-3)}^2+5^2\right]}^2={\left(D^2-6D+9+25\right)}^3={\left(D^2-6D+34\right)}^2$is the differential operator that annihilates the given function.