Classical Phonons: Solving Differential/Difference Equation

[aaaa202]

The below picture is from my book's derivation on the equations describing waves in matter. But problem is: I don't understand the solution of the differential equation - or "difference" equation (whatever that is). How is it solved with the proposed solution? If I plug it in I don't get anything meaningful.

 

[fzero]

If you plug it in, you get an equation that can be solved for the quantity $Ka$, which is presumably what the text did in the next paragraph. This type of solution is logical, we can rewrite the equation as

$$ \gamma u_s = u_{s+1} + u_{s-1},$$

so it is natural to look for solutions where each term has a common factor (possibly depending on $s$). The periodicity of the problem suggests that complex exponentials are relevant and they indeed have the property that solutions for different $s$ are multiples of one another.