Consider the following DDE: 

$ f''(x)+ f(x+ 1) = 0.$

For $ x \gg 1$, this DDE should approximately reduce to that of a simple harmonic oscillator: $f''(x)+f(x) \approx 0$. Suppose that for some physical reason I require $f(x) = \sin{x}$ for large $x$ and then use this as a ``history'' for the above DDE. I am then interested to see what $f(x)$ is in the vicinity of $x=0$. 

The problem is that (not surprisingly) depending on where I set the history condition (when using NDSolve), I get a different answer for $f(x)$ near $x=0$. Any suggestions for how to properly handle this?