I have a problem that I've boiled down to solving the difference equation $2(k-N) = j_{k+1} - 2j_k +j_{k-1}$, where $N$ is a constant.
I've been pouring through the site (http://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/sod/t), trying to figure it out, but I'm getting stuck.
I know that the solution to the homogeneous equation is $A + Bk$.
Then I try to find a solution to the non-homogeneous equation by guessing $j_k = a+bk$ and subsituting that into $j_{k+1} - 2j_k +j_{k-1} = 2k-2N$, getting $a(k+1)+b-2(ak_b) + a(k-1)+b = 2k - 2N$, i.e. $ak+a+b-2ak-2b+ak-a+b = 2k-2n$, so $\underbrace{(a-2a+a)}_{0}k - \underbrace{(a+b-2b-a+b)}_{0} = 2k - 2N$, so $k = N$, which doesn't make sense, so the guess must be wrong. But this seems to be the recommended technique.
I'd appreciate some help.