Solving a linear in-homogeneous constant coefficient recurrence relation

Question

I have the following homework problem:

a_n = a_n-1 + 12a_n-2 + 5(2)ⁿ

The problem states to find the general solution to the recurrence relation, as well as to find and solve for the particular solution.

I know how to solve for the homogeneous part (roots -3 & 4):

a_n^H = K₁(-3ⁿ) + K₂(4ⁿ)

Likewise, I know from my textbook that: a_n = a_n^H + a_n^P

However, I am stuck trying to find what the general form of the particular solution is. I understand that this should be a relatively easy problem, but I am completely lost and would appreciate any help that can be provided.

Answer 1

Hint:

WLOG set $a_m=b_m+c2^m$

Compare the coefficients of $2^n$ to find

$$5=c-c/2-12c/4\implies c=?$$