Linear Second Order ODE/y'' + 2 y' + 2 y = 0/Verification

Theorem
The equation:
 * $(1): \quad y = e^{-x} \paren {A \cos x + B \sin x}$

is a set of solutions to the second order ODE:


 * $y'' + 2 y' + 2 y = 0$

Proof
Differentiating $(1)$ twice $x$ gives:

Then:

thus demonstrating that $(1)$ is indeed a solution to the second order ODE given.