How do I solve the following system equations?
$x_i = 2x_{i-1} + 3x_{i-2}$, where $i = 1, 2, 3..., x_1 = 3$, and $x_2 = 6$.
The answer is $x_i = \frac{3}{4}(3^i - (-1)^i)$.
It's easy to solve:
$x_1 = 3, x_2 = 6, x_3 = 21, x_4 = 60...$, but what techniques should I use to derive the above generalized solution?
I found the answer, we need to use recurrence relation to solve it:
$r^2-2r-3=0, => r_1=-1,r_2=3$
then the solution is in the form of $x_i=α_1 r_1^i+α_2 r_2^i=α_1 (-1)^i+α_2 3^i$
by using two known initial values of $x_1=3$ and $x_2=6$, we can have $α_1=-3/4, α_2=3/4$, thus the solution is
$x_i=\frac{3}{4}(-(-1)^i+3^i$)