Find the general form of the solution to a linear homogeneous recurrence relation with constant coefficients for which the characteristic roots are 1,−2 and 3 with multiplicities 2,1 and 2, respectively. The relation also has a non-homogeneous part which is a linear combination of 3n and (−2) n .

Let us find the general form of the solution of a linear homogeneous recurrence relation with constant coefficients for which the characteristic roots are $1,−2$ and $3$ with multiplicities $2,1$ and $2$, respectively:

$a_n=B_1+B_2n+B_3(-2)^n+(B_4+B_5n)3^n$

Since the relation also has a non-homogeneous part which is a linear combination of $3n$ and $(-2)^n$, and $1,−2$ are characteristic roots with multiplicities $2,1$, respectively, then the partial solusion of a non-homogeneous equation is $n^2(C_1+C_2n)+C_3n(-2)^n.$