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May 22

All Solutions for F'(x) = F(x+a).

We know, for instance, that $\exp '(x)=\exp(x+0),$ $\sin 'x=\sin {\bigg (}x+{\frac {\pi }{2}}{\bigg )},$ and $\cos 'x=\cos {\bigg (}x+{\frac {\pi }{2}}{\bigg )}.$ But since all these are connected through Euler's formula, I was wondering whether there might not be other (non-exponential) functions with the same general property. — 79.118.167.195 (talk) 08:46, 22 May 2014 (UTC)[reply]

If a is fixed and not zero, then you can recover a solution by imposing basically aribitrary data in the form of a smooth function in [0,a] (with appropriate compatibility conditions at the end points) just by integration, I believe. Sławomir Biały (talk) 11:35, 22 May 2014 (UTC)[reply]

Note 'smooth' here has a precise definition, see smooth function. Dmcq (talk) 12:17, 22 May 2014 (UTC)[reply]

F(x) = 0 is a trivial solution for any a. Gandalf61 (talk) 12:26, 22 May 2014 (UTC)[reply]

Note that it suffices to consider a = 0 and a = 1. Count Iblis (talk) 14:45, 22 May 2014 (UTC)[reply]

You also have

\exp '(x)=\exp(x+2in\pi ),

for all integer n (including zero). The same function but more values for a. You can obviously add 2n

π

to

π

/2 in the trigonometric relations too. This suggests there are no more functions as the trigonometric and exponential functions satisfy F'(x) = F(x+a) for so many values.--JohnBlackburne^words_deeds 15:04, 22 May 2014 (UTC)[reply]

I don't really know anything about this equation, but it is a delay differential equation (linear with discrete delay), and that article contains a little bit of general information. —Kusma (t·c) 20:57, 22 May 2014 (UTC)[reply]