Is there a non trivial function such that
$$\frac{\partial f(x)}{\partial x} = f(x+1)$$
I'm not sure if the uniqueness of the ODE apply here because $\partial_x e^x =e^x$. If the continuous case did not work, what about the discrete series?
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If $f$ is a 1-variable function, then what is the point of using partial derivatives notation? – ShyamalSayak Dec 18 '23 at 16:20
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1See this answer of mine : https://math.stackexchange.com/questions/4611530/particular-differential-equation/4611566#4611566 – Abezhiko Dec 18 '23 at 16:20
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7Does this answer your question? How to solve differential equations of the form $f'(x) = f(x + a)$ – Hans Lundmark Dec 18 '23 at 16:51
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@Ryszard correct - you actually need $a=e^a$ which has complex solutions only – Conrad Dec 18 '23 at 19:47
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1check the Wikipedia of Delayed Differential Equations (DDEs) – Joako Dec 18 '23 at 23:54