How can I complexify the right hand side of this differential equation? I want to get a particular solution to the differential equation
$$
y''+2y'+2y=2e^x cos(x)
$$
and therefore I would like to 'complexify' the right hand side. This means that I want to write the right hand side as $q(x)e^{\alpha x}$ with $q(x)$ a polynomial. How is this possible?
The solution should be $(1/4)e^x(\sin(x)+\cos(x))$ but I cannot see that.

The point is that (for real $x$) $2 e^x \cos(x)$ is the real part of $2 e^x e^{ix} = 2 e^{(1+i)x}$.  Find a particular solution of $y'' + 2 y' + 2 y = 2 e^{(1+i)x}$, and its real part is a solution of $y'' + 2 y' + 2 y = 2 e^x \cos(x)$.
