Let $x(s)$ be a recursive equation defined for integers $s$ on $0 ≤ s ≤ t$ $$x(s) = x(s + 1)a + x(s − 1)b \text{ if } 0 < s < t$$ and boundary conditions $x(0) = u$, $x(t) = v$
Let $0<a<1$ and $a+b = 1$, and $t$ be a positive integer.
Show $x(s)$ is unique.
I noticed that the solution to the recursive equation is of the form $x(s) = \alpha + \beta\rho^s ,\rho=\frac{b}{a}$
Found that $$\alpha = \frac{v-u\rho^t}{1-\rho^t}$$ $$\beta = \frac{u-v}{1-\rho^t}$$
So the solution is of the form $$ x(s) = \frac{v-u\rho^t}{1-\rho^t} + \frac{u-v}{1-\rho^t}\rho^s $$
Is this the right approach? I'm not sure what to do next.