Solving the recurrence relation $x_{n+2}-5x_{n+1}+6x_{n}=5^{n}+n$

Question

SOLVE: $x_{n+2}-5x_{n+1}+6x_{n}=5^{n}+n$

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How to deal with it?

I use method: $r^2-5r+6r=0$ $x_{n}=A*2^{n}+B*3^{n}+..$

I can deal with similar problems: $x_{n+2}-5x_{n+1}+6x_{n}=5^{n}$ or $x_{n+2}-5x_{n+1}+6x_{n}=n$ but I have no idea for $x_{n+2}-5x_{n+1}+6x_{n}=5^{n}+n$

Answer 1

Note that the left side of your equation is linear in the $x_n$'s. Therefore, if you have two solutions, one for $\ldots=5^n $ and one for $\ldots=n $. Then the solution to the full problem is just the sum of the two solutions.