Question

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

7x − sin(x) + 20 cos(5x)

Answer 1

The differential operator $(D-\alpha)^n$ annihilates the functions:

$$e^{\alpha x},xe^{\alpha x},x^2e^{\alpha x}, ..., x^{n-1}e^{\alpha x}$$

Then the differential operator $D^2$ annihilates the function $x.$

We know that the differential operator $D^2-2\alpha D+(\alpha ^2+\beta ^2)$ annihilates $e^{\alpha x}\cos(\beta x)$ or $e^{\alpha x}\sin(\beta x).$

Thus the differential operator $D^2+1$ annihilates the function $\sin(x),$

the differential operator $D^2+25$ annihilates the function $\cos(5x).$

A linear differential operator that annihilates the given function is

$$D^2(D^2+1)(D^2+25)$$