$a[n + 2] - ea[n + 1] = 3f[n]; f[n] = e^n, a[-1] = \frac{e^2 - 3}{e^3}$
Wolfram Alpha gives the solution $a[n] = 3e^{n - 2}n + e^n$, but also says that there is no solution. The solution it gives does not seem to satisfy the difference equation.
My solution is $a[n] = e^{n - 1} - 3e^{n - 3} + 3ne^{n - 4}$, which seems to satisfy the difference equation. My solution does not even have the same number of terms as Wolfram's solution, and the two are not identical when graphed. Is Wolfram just bad at these, or am I doing something wrong?
$$a_{n+2} - ea_{n+1}=3e^n \implies \frac{a_{n+2}}{e^{n+2}} - \frac{a_{n+1}}{e^{n+1}} = 3e^{-2}$$
So $\frac{a_n}{e^n}$ is an arithmetic sequence, $$ \frac{a_n}{e^n}=(n+1)3e^{-2}+\frac{a_{-1}}{e^{-1}} \implies a_n=e^n+3ne^{n-2}. $$
Your solution gives $a_{n+2} - ea_{n+1}=3e^{n-2}$, a little off.
Wolfram Alpha is good but not always reliable.