Linear Non-homogeneous Recursive Relation

Question

this is the problem:

$f(n) = 4f(n-1)-3f(n-2)+2^n+n+3$ where $f(0)=1 , f(1)=4$

I know when the Non-homogeneous part is a product of an exponential function and a polynomial function like this:

$S^n(b_1n^k + b_2n^(k-1) + ... + b_k)$

the answer would be "the answer for homogeneous part" + $S^n(p_1n^k + p_2n^(k-1) + ... + p_k)$

since in this problem the case is "sum" of an exponential function and a polynomial function :

$2^n+n+3$

I don't know what to do! any Hints would be appreciated. thanks.

Answer 1

You have two pieces of that form. One piece is $2^n$ (pure exponential) and the other one is $1^n(n+3)$ (so you have a different exponential with base $1$). You can find particular solutions for both pieces of the non homogeneous term and add them up (this is possible because the equation is linear)