Classical Phonons: Solving Differential/Difference Equation

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The discussion focuses on solving differential and difference equations related to classical phonons in wave mechanics. The proposed solution involves rewriting the equation as $$ \gamma u_s = u_{s+1} + u_{s-1}$$, indicating a relationship between adjacent terms. The periodic nature of the problem suggests the use of complex exponentials, which provide solutions that are multiples of one another for different values of ##s##. The challenge lies in understanding how to derive meaningful solutions from the proposed equation.

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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.
 

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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.
 

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