Example 2-16 $a_{n}-a_{n-1}-6 a_{n-2}=3^{n}, \quad a_{0}=5, a_{1}=2$,
To find a particular solution using the method below, set the particular solution as $\alpha_{n}$ according to the recurrence relation,
$$
\begin{aligned}
\alpha_{n-1}-\alpha_{n-2}-6 \alpha_{n-3} & =3^{n \cdot 1} \\
3 \alpha_{n-1}-3 \alpha_{n-2}-18 \alpha_{n-3} & =3^{n} \\
\alpha_{n}-\alpha_{n-1}-6 \alpha_{n-2} & =3^{n}
\end{aligned}
$$

Therefore,
$$
\alpha_{n}-4 \alpha_{n-1}-3 \alpha_{n-2}+18 \alpha_{n-3}=0
$$
$\alpha_{n}$ satisfies a third-order homogeneous recurrence relation, whose characteristic function is
$$
\begin{array}{l}
x^{3}-4 x^{2}-3 x+18-(x+2)(x-3)^{2} \\
G(x)=\frac{P_{2}(x)}{(x+2)(x-3)^{2}}=\frac{A}{1+2 x}+\frac{B}{1-3 x}+\frac{C}{(1-3 x)^{2}} \\
\alpha_{n}=A(-2)^{2}+(B n+C) 3^{n}
\end{array}
$$

It can be seen that in this problem, a particular solution $\alpha_{n}=k n 3^{n}$ can also be taken. Substituting into the original non-homogeneous recurrence relation, we have
$$
\begin{array}{l}
k\left[n 3^{n}-(n-1) 3^{n-1}-6(n-2) 3^{n-2}\right]=3^{n} \\
k[9 n-3(n-1)-6(n-2)] 3^{n-2}=3^{n} \\
15 k=9, \quad k=\frac{3}{5}
\end{array}
$$

Thus, the particular solution is $\frac{3}{5} n 3^{n}$.
(2) For several non-homogeneous recurrence relations, the particular solution is
$$
a_{n}+c_{1} a_{n-1}+c_{2} a_{n-2}+\cdots+c_{k} a_{n-k}=r^{n} b(n)
$$

where $b(n)$ is a polynomial of degree $p$ in $n$, and $r$ is an $m$-fold root of the characteristic equation
$$
C(x)=x^{k}+c_{1} x^{k-1}+\cdots+c_{k} x+c_{k}
$$

The form of the particular solution is
$$
r^{n}\left[k_{0} n^{m}+k_{1} n^{m+1}+\cdots+k_{p} n^{m+p}\right]
$$

where $k_{0}, k_{1}, \cdots, k_{p}$ are undetermined constants, determined by the non-homogeneous recurrence relation.
If $r$ is not a root of $C(x)=0$, then let $m=0$.