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I try to solve for the specific function $f(x) = \frac{2-2a}{x-1} \int_0^{x-1} f(y) dy + af(x-1)$

It looks similar to the function used to find the Renyi's parking constant because it came out from a simple generalization of that problem.

The skill I have gained in my differential class can't even solve $f(x) = f'(x-1)$

I'm not looking for anyone to solve it. I just want to know the techniques for solving DE where functions and it's derivatives are evaluated at different points.(What's the terminology for this kind of DE?)

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    [Delay differential equation](http://en.wikipedia.org/wiki/Delay_differential_equation).2010-09-04
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    Can't $f(y + 1) = f'(y)$ be solved by just substituting $y = e^{ry}$ so you get $e^r = r$ (You need a special function to solve that). So then you appeal to some uniqueness and existence theorem. So the function that maps $y(t)$ to $y(t + 1)$ must be continuous and so must its derivative be.2010-09-04
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    Differential-difference equation. (Google it.)2010-09-04
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    The first one, naturally is a delay integro-differential equation (DIDE). ;)2010-09-04
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    whuber: My understanding of differential-difference equations is that they look something like this formula for the derivative of a Bessel function: $\frac{\mathrm{d}}{\mathrm{d}x}C_n(x)=\frac12(C_{n-1}(x)-C_{n+1}(x))$, and it can be shown that only the two solutions to the Bessel DE are the solutions to this differential-difference equation.2010-09-04
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    @J.M. : In your example the differences involve the subscript $n$. A *difference equation*, on the other hand, is essentially a recursion; it involves values of a function at the argument $x$ and discrete translates of $x$ itself such as $x-1$. A differential-difference equation includes derivatives and differences. See http://mathworld.wolfram.com/Difference-DifferentialEquation.html for example.2010-09-17
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    @whuber: I found http://books.google.com/books?id=5n2sN8rBU28C which corroborates your statement, though the usage I was accustomed to would be like the way the term was used in http://books.google.com/books?id=BUg4AAAAIAAJ&pg=PA192 and http://books.google.com/books?id=huuO6mKbVoEC&pg=PA214 . I suppose we are reading different books. :)2010-09-17

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