I have the following difference equation -
$2h_{x+1} - 5h + 2h_{x-1} = 0$ for $x = 1, 2, ...., 19$
The boundary conditions are $h_0 = 1$ and $h_{20} = 0$
How would I go about verifying that $h_x = At_1^x + Bt_2^x$ for $x = 0, 1, ..., 20$ is a general solution of this equation? And that $t_1$ and $t_2$ are roots of the quadratic equation $2t^2 - 5t + 2 = 0$?