How do we solve recurrence equations of the form:
$$ax_{n+1}+bx_n+cx_{n-1}=dn^p+e\;,$$ where $a,b,c,d,e$ are constants and $p\in \mathbb Z$?
Perhaps we could first solve the homogeneous equation $$ax_{n+1}+bx_n+cx_{n-1}=0\;.$$ Then we find the particular solution... but how? Guesswork?
Thanks.