I've made a few threads recently asking how to solve non-homogeneous recurrences and I think I've gotten the hang of it, but now I want to try a complicated thing like this:
$T(n) = 4T(n-1) + 2T(n-2) - 32T(n-3) + 59T(n-4) - 44T(n-5) + 12T(n-6) + 2^n + 5n^4 - 3n^3 + 2n + n \log(n)$
Where $T(0)... T(6) = 0...6$ for simplicity.
I constructed this in a very specific way.
So the homogeneous part has characteristic polynomial $x^6 - 4x^5 - 2x^4 + 32x^3 - 59x^2 + 44x - 12 = 0$ which is also $(x-2)^2 (x-1)^3 (x+3) = 0$
I assume this means the homogeneous part has solution of form:
$H(n) = \alpha_1 2^n + \alpha_2 n 2^n + \alpha_3 1^n + \alpha_4 n 1^n + \alpha_5 n^2 1^n + \alpha_6 (-3)^n$
Is this right?
Next, do I have to split up the non-homogeneous part into several pieces:
$2^n$ as its own piece
$5n^4 - 3n^3 + 2n$ as another piece
$n \log(n)$ as yet another piece?
Am I on the right track so far? My next question is how to correctly set up the "trial" equations for each piece.