I'm working on the problem below currently.
I feel that I am doing everything correctly, but I just have this tiny problem that's causing me issues! I've attached my working out below.
As tedious as it may be, it works out quite nicely in the end with A=4. So the next constant to find is B, which can be found by equating the terms in 'n'.
By doing so, we find that B=-3. which is nice. The problem is the -72B that is dangling at the end of the massive chunk of working out (just before I equated $4^{n-3}$).
I could find A because I could equate the $4^{n-3}$ and I could find B by equating $n$. However, I have this pesky $-72B$ that I can't get rid of or equate with anything.
What is going on here?
Also, the reason why I made the choice of particular solution $p_n=An\times4^n+Bn$ is because although normally I'd let it be $p_n=A \times 4^n + Bn$, a constant multiple of $4^n$ already exists in the general solution, so I cannot have another copy of it in the particular solution too.