Suppose we have $u'' + u' =0$ where $x \in (0,1)$ with the boundary conditions $u(0) =0, u(1) =1$.
We consider the BVP finite difference approximation $\frac{u_{j+1} -2u_{j} + u_{j-1}}{h^2} + \frac{u_{j+1} - u_{j-1}}{2h} = 0$ for $j=1, \ldots , (N-1)$. with $u_{0} = 0, u_{N} = 1$
Say $U_{j}$ is an approximation to $u(x_{j})$, where $x_{j} =jh$ with $h = \frac{1}{N}$.
is there some intuition to show that the solution is of the form $U_{j} = ar^j + b$ for some $a,b \in \Bbb{R}$ with $r \neq 1$ and $r$ satisfying $(2+h)r^2 - 4r + (2-h) = 0$.