$\displaystyle f_{n+2} - 2f_{n+1}+ 5f_n = 0$

given that $\displaystyle f_0=1$ and $\displaystyle f_1=3$

Auxiliary equation:

$\displaystyle m^2 - 2m + 5 = 0$

Therefore:

$\displaystyle m = 1 + 2i$ or $\displaystyle m = 1 - 2i$

What would the complementry fuction be and paticular solution to work towards the solution of:

$\displaystyle f_n = [(1-i)/2][1+2i]^n + [(1-i)/2][1-2i]^n$