Uniqueness of a solution class for a homogeneous recurrence equation

Question

I have a homogeneous recurrence equation.

$x_i = (1-p)x_{i-1} + px_{i+1}, p \in (0,1)$

Using the classical methods (solving the characteristic polynomial) I can show that

if $p=1/2$, then the sequences described by $x_i = c_1+c_2i$ are solutions $\forall c_1,c_2 \in \mathbb R$.

if $p \ne 1/2$, then the sequences described by $x_i = c_1+c_2(\frac{1-p}{p})^i$ are solutions $\forall c_1,c_2 \in \mathbb R$

According to a theorem here (point 2) there is no other solutions.

I am now asked to explain why there are no other solutions. Is the only explanation a reference to the theorem/ its proof or there is a straightforward argument why in my case there could be no other solutions?

Answer 1

$c_1$ and $c_2$ are determined by two given $x_i$ and $x_k$ (for different indices). Suppose now that there is another solution $y$ not identical to the initial solution $x$ but which also satisfies the given xs. The difference of this other solution and your initial solution, write it as $d_l=x_l-y_l$, solves the difference equation as well with $d_i=d_k=0$. If $d_{i\pm 1}=0$ then the difference equation gives, successively, that all $d_l$s are zero, contradicting that $x$ and $y$ are not identical. So the $d_{i\pm 1}$ elements have to be different from zero. But then this non zero $d$ is then propagated by the difference equation and we get $d_k\neq 0$, a contradiction again.