I've tried to find a solution to my problem, but I didn't managed to. The thing is about non-homogenous linear equations. I've got a linear equation in the form of $a_n = 3 a_{n-1} + 50 a_{n-2} + 12 a_{n-3} + 216 a_{n-4} + 8 \cdot 9^n + 6n \cdot 9^n $.
As far as I know, I can find a solution of associated homogenous linear equation, and then find solutions for others parts of problem. I found a general solution for associated homogenous linear equation, I know how to work with terms like a $8 \cdot 9^n$, but I don't have any idea how to deal with $6n \cdot 9^n$. I've searched on the internet any possible ways to deal with things like this, but I've got no answer. Can you help me?
I prefer to rewrite the equation as $$a_n = 3 a_{n-1} + 50 a_{n-2} + 12 a_{n-3} + 216 a_{n-4} + (8 + 6n)\,9^n$$ Solving the homogeneous equation, you noticed that $9$ is a root of the characteristic equation and since the last term is of first degree, then the general solution is of second degree.
So, let, for the particular solution, $a_n=(\alpha+\beta n +\gamma n^2)\,9^n$; replace and group terms to end with $$(425 \beta -845 \gamma -1944)+(850 \gamma -1458) n=0$$ making $\gamma=\frac{729}{425}$ and then $\beta=\frac{288441}{36125}$ and the $\alpha$ term will be added to the coefficient of the $9^n$ term which already appeared in the solution of the homogeneous equation.