Finding particular solution of a difference equation

Question

How Can I find a general form of a particular solution of such an equation:

                 []−0.1[−1]−0.56[−2]=3[]+2[−1]

[]=$0.5^n$⁡[]

The system is initially at rest and causal.

Answer 1

Taking the $z$ transform, the system function is $$H(z)=\frac{Y(z)}{X(z)}=\frac{3+2z^{-1}}{1-0.1z^{-1}...}$$

The particular solution is the response of the system to the given input $x[n]$, whose $z$-transform is can be easily calculated. The $z$-transform of the response is given by $$Y(z)=H(z)X(z)$$ Now just set the initial conditions to zero and using partial fraction expansion and inverse $z$-transform $y[n]$ can be found.