According to my textbook, there's one particular category of inhomogeneous linear difference equation which takes the form of:
$Y_{t+1} - aY_t = P_{n}(t)$
where $P$ is a polynomial function of degree $n$.
And we could solve for the general solution depending on the value of $a$.
First let $y^{*}$ be the particular polynomial function of degree $m$ and substitute it back into the original formulae.
If a $\neq$ 1, we could then deduce that $m = n$, which is $y^{*} = Q_{n}(t)$ and solve it accordingly. ($Q_{n}(t)$ being a polynomial function of degree $n$)
If a $=$ 1, we could then deduce that $y^{*} = tQ_{n}(t)$.
I didn't really get why we could reach above conclusions.
I have tried transform the original formulae into:
$Y_{t+1} - aY_{t} =Y_{t+1}-Y_{t} + (1-a)Y_{t} =\triangle Y_{t} +(1-a)Y_{t}=P_{n}(t)$
I still couldn't quite get it why.
Could someone please point out what I have missed?