Question

find linearly independent function that are annihilated by the given differential operator D⁵,(D²+4D)

Answer 1

$$Solution$$

The linearly differential operator $D^5(D^2+4D)$ which is $D^6(D+4)$ in the form $D^{n+1}(D +\alpha )$. Here $n=5$ and hence it can annihilate the independent functions that are in the form $x^{n \le5}$ and $e^{-\alpha x}$, then we can have the functions being annihilated by the given differential operator as $x^0, x^1, x^2, x^3, x^4,x^5$ and $e^{-4x}$.

$\therefore 1, x, x^2, x^3, x^4,x^5\ and\ e^{-4x}$

Where $n=0,1,2,3,4,5\ and\ \alpha=4$