I have recurrence relation $x_{n+4}-3x_{n+3}+4x_{n+2}-8x_{n}=0,x_0=0$.
Here I tried to solve it using the characteristic equation method. I got roots $t=-1,t=2$.
General solution looks like this $a_n=(-1)^nC_1+2^nC_2$ but how can I find $C_1$ and $C_2$?
I tried to write general solution using the initial condition $a_0=C_1+C_2=0$ but it didn't work, since I can't compose a system for further solution with only one initial condition.
Can I find $a_1$ somehow to build system to find $C_1$ and $C_2$?
Best Answer
So actually, it is more complicated than that. If you put in $x_n = a^n$ and then divide through by $a^n$, you will get $$ a^4-3a^3+4a^2-8 = 0, $$ which indeed is solved by $-1,2$ as you claim, but also has 2 complex solutions $1 \pm i\sqrt{3}$, so the general form of the solution would look something like $$ x_n = A(-1)^n + B 2^n + C 2^n e^{-i\pi n/3} + D2^n e^{i\pi n/3} $$ so you would need 4 initial conditions to determine the recurrence completely.
This is in general the rule: $n$-th degree recurrences require $n$ initial conditions to be fully determined.