It makes no sense to ‘put $a_n=b_n$’ when you have no $b_n$ in the problem. What you wanted is the auxiliary equation $b^2-6b+9=0$, which you correctly solved to find the general solution
for some constants $c_1$ and $c_2$. You determine those by using the known values of $a_0$ and $a_1$: when $n=0$ equation $(1)$ becomes
and when $n=1$ it becomes
Now simplify $(2)$ and $(3)$ and solve for $c_1$ and $c_2$ to finish deriving the solution.
To verify it by mathematical induction you must do three things:
- verify that $3^0+0\cdot3^0=a_0$, i.e., that the formula gives the correct value when $n=0$;
- verify that $3^1+1\cdot3^1=a_1$, i.e., that the formula gives the correct value when $n=0$; and
- show that if $n\ge 2$ and $a_k=3^k+k3^k$ for $k=0,1,\ldots,n-1$, then $a_n=3^n+n3^n$.
(1) and (2) are the basis steps of your induction, and (3) is the induction step: in it you’re showing that if the expression $3^k+k3^k$ gives the right value for $a_k$ for all $k<n$, then it also gives the correct value for $a_n$. In carrying out this step you’ll use the recurrence that defines the sequence.