non-homogeneous Recurrence Relation for f(x) = n^2

Question

Im having some trouble with a non-homogeneous Recurrence Relation. My question is:

$u_{n} - 5u_{n-1} + 4u_{n-2} = n^2$

My working out so far:

$r^{2}-5r+4r = 0$ = (r-1)(r-4)

Giving the roots 1 and 4 so my homogeneous equation is:

(λ + 4β)

My issue is that I'm not sure how to guess the non-homogeneous particular, I've read if it equals $n^2$ the guess should be of the form $A_{2}n^{2} + A_{1}n + A_{0}$ but I'm not sure how to implement this.

Any help would be appreciated.

Answer 1

Guess $$u_n=a + bn+c n^2+d n^3$$ and replace in $$u_{n} - 5u_{n-1} + 4u_{n-2} -n^2=0$$ You should arrive to something like $$(-3 b-27 d+11c)+ (33 d-6 c)n-(9 d+1) n^2=0$$ I am sure that you can take it from here.