= Solution
Divide the <second-order difference equation> by $h^2$ and set $x=nh$. The first quotient is a <central finite difference> approximation to $y''(x)$; the quotient $(y_{n+1}-y_{n-1})/(2h)$ approximates $y'(x)$. Their <Taylor series> errors are $O(h^2)$ for a sufficiently smooth function. This is therefore a <finite difference method> for the <second-order linear differential equation> in part (a).
At a fixed $x\geq0$, take $h\to0$ with $n=x/h$ integral. Since $\log r_-=-2h+O(h^3)$, the exact bounded discrete solution satisfies
$$
\boxed{r_-^{x/h}=\exp\bigl(-2x+O(xh^2)\bigr)\longrightarrow e^{-2x}.}
$$
Likewise $(1-2h)^{x/h}\to e^{-2x}$. The decay selected by boundedness in the <linear recurrence relation> becomes exactly the decay selected by boundedness in the <ordinary differential equation>.
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