how would I go about tackling this kind of question? I'm struggling with the concept of recurrence relations. Any advice would be apreciated.
Find the general solution of each of the following recurrences.
(a) $u_n = 7u_{n−1} − 12u_{n−2}$
(b) $u_n − 3u_{n−1} = 4n^2$
(c) $u_n − 3u_{n−1} = 5 \cdot 4^n$
Hints:
(a) Try solutions of the form $u_n=a \lambda^n$.
(b) Try solutions of the form $u_n = a 3^n + b n^2 + c n + d$
(c) Try solutions of the form $u_n = a 3^n + b 4^n$
The main principle is to combine solutions of the homogenous equation with a particular solution.
(a) is a homogenous equation and you can just try $u_n=a \lambda^n$ and see which $\lambda^n$ work.
(b) and (c) are not homogenous but their homogenous part is easy. For the particular solution for the non homogenous part, just try something of the same form.