Question

Find a linear differential operator that annihilates the given function. (Use D for the differential operator.) 4 ex cos 3x

Answer 1

If the function is

`y = 4 e^x \cos(3x)`

then we have derivatives

`Dy = 4 e^x \cos(3x) - 12 e^x \sin(3x)`

`D^2y = -32 e^x \cos(3x) - 24 e^x \sin(3x)`

Now consider the linear ODE

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

Substituting
`y` and its derivatives reduces the equation to

`(-32a + 4b + 4c) \cos(3x) + (-24a - 12b) \sin(3x) = 0`

Now,

`-24a - 12b = 0 \implies b = -2a`

`-32a + 4b + 4c = 0 \implies c = 10a`

Then the minimal ODE with the given solution is

`aD^2y -2a Dy + 10ay = 0`

or

`\boxed{D^2y  -2 Dy + 10y = 0}`