heterogeneous recurrence with f(n) as constant How to solve this $s_{n+1}=4s_{n-1}-3s_n+5$ where f(n)=5 conditions $s_0=-3$ $s_1=3$
I calculated the general solution $s_n=c_1*(-4)^n+c_2*1^n$ of this recurrence. The roots are $q_1=-4$ and $q_2=1$ but I have problem with particular solution with method of prediction .
I have problem with this 5 as constant
solution(homo+heterogeneous) is $s(n) = n - (-4)^n - 2$
I learn in Polish so I can't find too much words to describe this

$$s_{n + 2} = -3~s_{n + 1} + 4~s_{n} + 5$$
$$s_{n + 3} = -3~s_{n + 2} + 4~s_{n + 1} + 5$$
So
$$s_{n + 3} - s_{n + 2} = (-3~s_{n + 2} + 4~s_{n + 1} + 5) - (-3~s_{n + 1} + 4~s_{n} + 5)$$
$$s_{n + 3} = -2~s_{n + 2} + 7~s_{n + 1}$$
The roots are $-4$, $1$ and $1$, so the final form is:
$$s_n = A~(-4)^n + B~1^n + C~n~1^n$$
$$s_n = A~(-4)^n + B + C~n$$
Use initial conditions to find $s_0$, $s_1$, and $s_2$ to find $A$, $B$, and $C$.
