Let a linear, time-invariant, causal discrete linear system be described by the following difference equation:
$$y[n] = 0.9y[n-1] + 0.1x[n]$$
Assuming that $y[-1]=2$ and $x[n]=20\cos(\Omega n)u[n]$, $\Omega = \omega T_s = 0.2\pi$, find the total response, identifiying the terms related to the zero state and zero input responses.
I've found the transfer function of the system to be: $$H(z) = 0.1\frac{z}{z-0.9}$$
and from it I've determined the magnitude and phase of the steady-state response: $$|H \left(e^{j \Omega}\right)| = \frac{0.1}{\sqrt{1.81-1.8\cos(\Omega)}}$$ $$\angle H \left(e^{j \Omega}\right) = -\arctan\left[\frac{-\sin(\Omega)}{1-0.9\cos(\Omega)} \right]$$
Therefore the steady-state response of the system is: $$y_{ss}[n] = 20 \frac{0.1}{\sqrt{1.81-1.8\cos(\Omega)}} \cos\left(\Omega n -\arctan\left[\frac{-\sin(\Omega)}{1-0.9\cos(\Omega)} \right] \right)$$
$$\therefore \boxed{y_{ss}[n] = 3.363\cos(0.2\pi n + 1.138)}$$
But how do I find the total response (zero state and zero input) from the steady-state response?