Annihilator Method. Getting wrong answer

Question

I am doing by this method

Annihilate this function: $(7x^2+5)e^{2x}.$

I am getting the answer $(D^2-2)^5$, but this doesn't annihilate the function.

What am I doing wrong?

Answer 1

You have trivial computation errors, $$ D^2-4D+4=(D-2)^2 $$ and $$ (x^a)^b=x^{a·b}. $$ But you are starting from the wrong general formula, if there are no trigonometric functions involved, you do not use the quadratic polynomial, for $x^me^{\lambda x}$ the annihilator is $$ (D-λ)^{m+1} $$

The reason for the difference is that in the trigonometric case the differential operator annihilates two exponential terms with complex conjugate exponential factors at once. Then the annihilator is the product of the two corresponding exponential annihilators, $$ (D-(\alpha+i\beta))^{m+1}(D-(\alpha-i\beta))^{m+1} =((D-α)^2+β^2)^{m+1}=(D^2-2αD+α^2+β^2)^{m+1}. $$