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I feel like this should be really easy, but how do I solve $f'(x+1)=f(x)$? I am extremely new to differential equations, and can't figure out what I'm missing. Most resources that I look at are in terms of $y''+y'=e^x$ or whatever, and none that I've seen discuss composition of functions. WolframAlpha can't figure out what I'm telling it; it doesn't seem to think that there's anything to solve.

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It is a delay defferential equation. For that particular equation, you can fix the value of the solution on any interval of length $1$, say $f(x)=\phi(x)$ for $x\in[-1,0]$, and then construc the solution on $[0,1]$ as follows: $$ f'(x)=f(x-1)=\phi(x-1)\text{ if }0\le x\le1\implies f(x)=\phi(0)+\int_0^x\phi(t-1)\,dt,\quad 0\le x\le1. $$ Now you can find $f$ on $[1,2]$ and so on.

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