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Here's an excerpt of a lecture note I am reading (I've highlighted the beginning of the part that I don't understand):

enter image description here

I don't understand how derivative comes into the picture.

Here's some context on my situation:

My Algebra course has covered (very briefly) non-homogenous second order difference equation. I want to understand to intelligently make a guess when finding a particular solution instead of memorizing all the possible forms. Nowhere during my lecture was derivative mentioned so I am really confuse about this point. Also, I have not yet covered ordinary differential equations, so if the answer is dependent on knowing ordinary differential equations, please mention which ideas that it specifically making reference to.

Did
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mauna
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  • This refers to DIFFERENTIAL equations. Is your question about how to apply this to DIFFERENCE equations? – Did Dec 24 '13 at 09:13
  • @Did Yes. That's exactly what I wanted to know. – mauna Dec 24 '13 at 10:02
  • Then the highlighted part is irrelevant to your question. – Did Dec 24 '13 at 11:21
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    @Did, the differences (or shifts) of "commonly encountered functions" are mostly of the same forms. So the highlighted part is relevant, but translated for differences. – vonbrand Aug 18 '15 at 17:28
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    @vonbrand Once again: the text above, in particular its highlighted part, are concerned about differentials and express the fruitful idea that the differentials of "usual" functions are (often) usual functions themselves. The OP was asking about difference equations, that is, 1. a topic which is not the subject of the cited text, and 2. a setting where the idea mentioned above, while still true, becomes much less interesting. – Did Aug 18 '15 at 21:26
  • @Did, derivatives aren't relevant; their finite counterparts sure are. – vonbrand Aug 20 '15 at 17:55
  • @vonbrand If you say so. And the link with the question is? – Did Aug 20 '15 at 20:00

1 Answers1

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The heuristics is to compare the behaviour of the functions $t\mapsto x(t)$ solving the differential equation $x'(t)=F(x(t))$ to the sequences $(x_n)$ solving the difference equation $x_{n+1}=x_n+F(x_n)$. If $x(0)=x_0$, one can hope that $x(t)$ at time $t=n$ stays close to $x_n$, at least for values of $n$ that are not too large. This hope is based on the observation that, by definition of the derivative $x'(t)$, $x(t+s)=x(t)+F(x(t))s+o(s)$ when $s\to0$, which suggests that $x(t+1)$ might be close to $x(t)+F(x(t))$. Naturally, these are only approximations, whose quality may worsen when $t$ or $n$ become large.

A general remark is that while the differential equation $x'(t)=F(x(t))$ is often exactly solvable using usual functions, the difference equations $x_{n+1}=G(x_n)$ very seldom are, hence, despite the warning above about accumulating errors, it is often a good strategy to study the former for $G:\xi\mapsto F(\xi)+\xi$ to get some information (even if only qualitative) about the latter for $F$.

Did
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