Question

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

Answer 1

Let
`y=\cos(2x)`, with first two derivatives

`Dy = -2\sin(2x)`

`D^2y = -4\cos(2x)`

Consider the linear ODE

`aD^2y + bDy + cy = 0`

Substitute
`y` and its derivatives and solve for the unknown coefficients
`a,b,c`.

`-4a\cos(2x) - 2b\sin(2x) + c\cos(2x) = 0`

`\implies \begin{cases}-4a+c=0 \\ -2b=0 \end{cases}`

`\implies b=0 \text{ and } c= 4a`

Then the minimal ODE with this solution is

`aD^2y + 4ay = 0 \implies \boxed{D^2y + 4y = 0}`