I want to ask for a hint in solving the following problem from Berkeley Problems in Mathematics:
Let $h>0$ be given. Consider the linear difference equation $\frac{y((n+2)h)-2y((n+1)h)+y(nh)}{h^{2}}=-y(nh),n\in \mathbb{Z}^{*}$ 1) Find the general solution of the equation by trying suitable exponential subsititions.
2) Find solutions with $y(0)=0$ and $y(h)=h$. Denote it by $S_{h}(nh)$.
3). Let $x$ be fixed and $h=\frac{x}{n}$. Show that $\lim_{n\rightarrow \infty}S_{\frac{x}{n}}(\frac{nx}{n})=\sin[x]$
I am having trouble even using the first step. I tried $y=e^{P(t)}Q(t)$ but I feel it is unlikely to generate the required relation. So I am stuck. I think I need a hint as the book does not have a solution to this problem.