Question

1. Find a linear differential operator that annihilates the given function. (Use D for the differential operator.)

e−x + 5xex − x2ex

2. Find a linear differential operator that annihilates the given function. (Use D for the differential operator.)

cos 6x

Answer 1

 1)A linear differential operator g(D) is called annihilates the function f(x) iff $g(f(x)) = 0$ where $D = \frac{d}{dx}$ .

Given $f(x)=e^{−x} + 5xe^x − x^2e^x$ _____________(1)

We known $(D-A)^n$ annihilates$C_1e^{Ax}+C_2xe^{Ax}+C_3x^2e^{Ax}+ …+ C_{n-1}x^{n-1}e^{Ax}$ .

Hence, $(D+1)$ annihilates $e^{-x}$ and $(D-1)^3$ annihilates $5xe^x-x^2e^x$ .

Hence, $(D+1)(D-1)^3$ annihilates $f(x)$ .

2) Given$f(x) = cos(6x)$ .

$\implies Df = -6sin(6x) \implies D^2f = -36 cos(6x) \\ \implies (D^2+36)f(x) = 0$

Thus, $(D^2+36)$ annihilates $cos(6x)$ .