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

$\boxed{D^2y -2 Dy + 10y = 0}$