How to find a particular solution for $a_{n}-a_{n-1}-6a_{n-2}=5\times3^n$

Question

I tried $a_{n} = c \times 3^n$ and I got $c = 3$. So $a_{n} = 3 \times 3^n$ is a particular solution.

But it is a solution in the homogeneous form already. How to find another solution that's not a homogeneous solution?

Answer 1

You should have gotten $c-c/3-(6/9)c=5$, or $0=5$.

Whenever the first guess is already in the homogeneous solution, multiply it by n, Try $a_n=c\times n3^n$