The homogeneous equation is obtained by putting all the inputs x(n)=0 (same as
saying all 
). Notice that this corresponds to allowing the system to relax
from some initial condition with no inputs driving it.
Method
Divide through by 
and write the homogeneous equation as :
Substitute a trial solution 
.
Now 
because this gives only the uninteresting solution y(n)=0 .
This is called the characteristic equation (CE). The N roots of the CE are called
the characteristic roots. The characteristic roots tell us about the natural response
of the system when we give it a bit of a kick but no continuing driving inputs.
If the roots are 
the factored form of the CE is :
The roots are real or complex numbers. For real 
, when complex the roots occur
in complex conjugate pairs. The solution to the homogeneous equation is thus of the form :
Where the N unknown constants 
are determined from N initial conditions.
We write 
here to emphasize that this is not a general solution to the
original difference equation. It is an initial condition solution for 
and the initial conditions are given by values of 
.
etc
are a set of N linear algebraic equations to determine the unknown constants from
the initial conditions.
Case of multiple roots
If the CE contains multiple roots, eg a factor of 
, terms of the form :
will appear in the solution.
Test for system stability
The solution to the homogeneous equation represents the response of the system with
no driving inputs. We just have it in some initial state and watch the output decay.
Clearly this response should decay to zero as 
. This happens if and only
if 
for all k (since there is a term 
in the solution).
Example
A causal LTI system is described by the DE :
Find the IC response to the intial conditions :
y(-1)=1 and y(-2)=0
Solution
The CE is :
So the roots are 
and 
.
Thus the IC response is of the form :
and 
are determined from the initial conditions :
Which can be solved by elimination to give 
and 
Note that since both roots have magnitudes <1 the system is stable.
Sample solutions :
etc.