I have the difference equation $$\frac{\epsilon}{h^2}(U_{m+1} - 2U_{m} + U_{m-1}) + \frac{1}{h}(U_{m+1} - U_{m-1}) = 0$$ for $0<x<1$, $\epsilon >0$, and $U_0 = \alpha, U_{M}=\beta$ and $m = 1,\cdots, M$
I see that $U_m = 1$ is indeed a solution to the difference equation, but why is $U_m = (\frac{\epsilon + h}{\epsilon - h})^m$? I have plugged this in and don't see any cancellation.
Let $a=\frac \epsilon {h^2}$ and $b=\frac 1h$ this gives $$(a+b)U_{m+1}-2aU_m+(a-b)U_{m-1}=0$$So, the characteristic equation is $$(a+b)r^2-2ar+(a-b)=0$$ the roots of which being $r=1$ and $r=\frac{a-b}{a+b}$
So, before unsing any condition, the general solution is $$U_m=c_1+c_2\left(\frac{a-b}{a+b}\right)^m=c_1+c_2\left(\frac{\epsilon -h}{\epsilon+h }\right)^m=c_1+c_3\left(\frac{h-\epsilon }{h+\epsilon }\right)^m$$ which satisfies the equation.
But the solution you worked with does not.
Probably, one more typo in a textbook.