In my mind, the steps are
- 1: Find the homogeneous solution using the characteristic equation
- 2: Use "undetermined coefficients" method of finding the particular solution.
However, what is the form of the particular solution for these kinds of equations?
I am trying to find the general form of $x[n]$: $$ x[n] + x[n-1] = {\delta}[n] $$
Is there no way to solve this using the Z-transform?
You can do something like that, 1.) $x[n]=(-1)^n$ is a homogeneous solution to your example problem, to 2.) find via variation of coefficients $$ x[n]=(-1)\cdot y[n]\implies y[n]-y[n-1]=\delta[n], $$ using that the delta sequence is zero almost everywhere, so that the alternating sign change does not influence it.
Now if you set $y[n]=0$ for $n<0$, then if follows that $y[n]=1$ for $n\ge 0$, as the recursion tells you that it is the previous value except at $n=0$, where $1$ is added.
The method of undetermined coefficients is not applicable here, as it requires that the right side is composed of terms $r[n]=p(n)q^n$ for some $q\ne0$ and some polynomial $p(n)$.