Answer to Question #188494 in Discrete Mathematics for Amit

Question

1-Show that u_x=c₁e^ax +c₂e^-ax solution of the difference equation

u_x+1-2u_xcosha+u_x-1

Answer 1

$u(x+1)+u(x-1)=c_1 e^{a(x+1)} +c_2 e^{(-a(x+1) )}+c_1 e^{a(x-1)} +c_2 e^{(-a(x-1))}$

$u(x+1)+u(x-1)= (c_1 e^{a(x+1)} +c_1 e^{a(x-1)} )+(c_2 e^{(-a(x+1) )}+c_2 e^{(-a(x-1) )} )$

$\Rightarrow (e^a+e^{(-a)} ) c_1 e^{ax}+(e^{(-a)}+e^a ) c_2 e^{(-ax)}\\\Rightarrow (e^a+e^{(-a)} )(c_1 e^{ax}+c_2 e^{(-ax)} )\\\Rightarrow 2coshau(x)$

This equation holds for all $x\in R$